On the Operators with Numerical Range in an Ellipse

We give new necessary and sufficient conditions for the numerical range $W(T)$ of an operator $T \in \mathcal{B}(\mathcal{H})$ to be a subset of the closed elliptical set $K_\delta \subseteq \mathbb{C}$ given by \[ K_\delta {\stackrel{\rm def}{=}} \left\{x+iy: \frac{x^2}{(1+\delta)^2} + \frac{y^2}{(1-\delta)^2} \leq 1\right\}, \] where $0<\delta<1$. Here $\mathcal{B}(\mathcal{H})$ denotes the collection of bounded linear operators on a Hilbert space $\mathcal{H}$. Central to our efforts is a direct generalization of Berger's well-known criterion for an operator to have numerical radius at most one, his so-called strange dilation theorem. We next generalize the lemma of Sarason that describes power dilations in terms of semi-invariant subspaces to operators $T$ that satisfy appropriate dilation properties. This generalization yields a characterization of the operators $T\in \mathcal{B}(\mathcal{H})$ such that $W(T)$ is contained in $K_\delta$ in terms of certain structured contractions that act on $\mathcal{H} \oplus \mathcal{H}$. As a corollary of our results we extend Ando's parametrization of operators having numerical range in a disc to those $T$ such that $W(T)\subseteq K_\delta$. We prove that, if $T$ acts on a finite-dimensional Hilbert space $\mathcal{H}$, then $W(T)\subseteq K_\delta$ if and only if there exist a pair of contractions $A,B \in \mathcal{B}(\mathcal{H})$ such that $A$ is self-adjoint and \[ T=2\sqrt\delta A + (1-\delta)\sqrt{{1+A}}\ B\sqrt{{1-A}}. \] We also obtain a formula for the B. and F. Delyon calcular norm of an analytic function on the inside of an ellipse in terms of the extremal $H^\infty$-extension problem for analytic functions defined on a slice of the symmetrized bidisc.

= x + iy : x 2 (1 + δ) 2 + y 2 (1 − δ) 2 ≤ 1 , where 0 < δ < 1.Here B(H) denotes the collection of bounded linear operators on a Hilbert space H. Central to our efforts is a direct generalization of Berger's well-known criterion for an operator to have numerical radius at most one, his so-called strange dilation theorem.Specifically, we show that, if T acts on a finite-dimensional Hilbert space H and satisfies a certain genericity assumption, then W (T ) ⊆ K δ if and only if there exists a Hilbert space K ⊇ H, operators X 1 and X 2 on H and a unitary operator U acting on K such that ) where P H denotes the orthogonal projection from K to H.
We next generalize the lemma of Sarason that describes power dilations in terms of semi-invariant subspaces to operators T that satisfy the relations (0.1) and (0.2).This generalization yields a characterization of the operators T ∈ B(H) such that W (T ) is contained in K δ in terms of certain structured contractions that act on H ⊕ H.
As a corollary of our results we extend Ando's parametrization of operators having numerical range in a disc to those T such that W (T ) ⊆ K δ .We prove that, if T acts on a finite-dimensional Hilbert space H, then W (T ) ⊆ K δ if and only if there exist a pair of contractions A, B ∈ B(H) such that A is self-adjoint and We also obtain a formula for the B. and F. Delyon calcular norm of an analytic function on the inside of an ellipse in terms of the extremal H ∞ -extension problem for analytic functions defined on a slice of the symmetrized bidisc.In this section we shall introduce notation, recall some classical results from the literature and describe the results of the paper.
For H a complex Hilbert space, B(H) will denote the collection of bounded linear operators on H.For T ∈ B(H), σ(T ) will denote the spectrum of T , and W (T ) the numerical range or field of values of T , which is defined by the formula Accounts of the well-established theory of the numerical range of a bounded linear operator can be found in the books [12,18].
In this paper, for fixed δ ∈ [0, 1), we study operators whose numerical ranges lie in the closed subset K δ of the complex plane C bounded by the ellipse with major axis [−1 − δ, 1 + δ] and minor axis i[−1 + δ, 1 − δ], so that (1.1) Note that K 0 = D − , the closure of the unit disc D = {z ∈ C : |z| < 1} in C. In this case there is already a well-developed theory, due to Berger [6], Ando [4], Kato [15], Dritschel and Woerdeman [10] and others, of operators whose numerical ranges are contained in D − .
Theorem 1.2.(Berger's Theorem) If T ∈ B(H), then W (T ) ⊆ D − if and only if there exists a Hilbert space K ⊇ H and a unitary operator U ∈ B(K) such that where P H denotes the orthogonal projection from K to H.
As the Sz-Nagy Dilation Theorem [21] asserted that if T ∈ B(H), then T ≤ 1 if and only if there exists a Hilbert space K ⊇ H and a unitary operator U ∈ B(K) such that Berger referred to his theorem as a strange dilation theorem.
Here, we shall show how Berger's theorem can be generalized to give conditions for the numerical range of an operator to be a subset of the elliptical set K δ .To do this, we reinterpret the formulae (1.3) as a representation theorem for a certain rational B(H)valued function defined in terms of T .Firstly, observe that if in Theorem 1.2 we relax the condition that H be a subspace of K to the condition that H merely be identified with a subspace of K via a Hilbert space isometry I, then the formula (1.3) takes the form Secondly, observe that the formulae (1.5) imply that, for all z ∈ D, and for any unitary operator U, The above observations give us an alternative formulation of Berger's strange dilation theorem.
Theorem 1.6.(Reformulation of Berger's Theorem) Let T ∈ B(H) be an operator satisfying σ(T ) ⊆ D − .Then W (T ) ⊆ D − if and only if there exists a Hilbert space K, an isometry I : H → K, and a unitary operator U ∈ B(K) such that 1 −1 2 zT 1 − zT = I * 1 1 − zU I for all z ∈ D. (1.7)

A generalization of Berger's strange dilation theorem
The following result, which we prove in Section 3, is a Berger theorem on an ellipse, modelled on the alternative formulation in Theorem 1.6.
Theorem 1.8.For an operator T ∈ B(H) and for any δ ∈ [0, 1), the numerical range W (T ) is contained in the closed set K δ if and only if there exists a Hilbert space K, an isometry I : H → K, and a unitary operator U ∈ B(K) such that 1 − 1 2 zT 1 − zT + δz2 = I * 1 1 − zU I for all z ∈ D.
(1.9) Theorem 1.8 prompts us to say that, for an operator T ∈ B(H) satisfying σ(T ) ⊆ K δ , a triple (K, I, U) is a strange dilation of T relative to K δ if K is a Hilbert space, I : H → K is an isometry, U is a unitary operator on K and the formula (1.9) holds.
For any δ ∈ (0, 1), an alternative expression of the notion of a strange dilation of T relative to K δ is in terms of a pair (X 1 , X 2 ) of operators such that σ(X 1 ) ∩ σ(X 2 ) = ∅, X 1 + X 2 = T and X 1 X 2 = δ.Such a pair exists whenever T is generic 1 for K δ .
Given such a pair (X 1 , X 2 ) we may express the left hand side of equation (1.9) in the definition of a strange dilation via partial fractions, so that equation (1.9) becomes which is an interesting modification of the relations (1.3).
1.3.The Sarason segue and even stranger dilations Consider the operators T ∈ B(H) that satisfy a simpler condition than that of Theorem 1.8, to wit, that there exists a Hilbert space K ⊇ H, an isometry I : H → K, and a unitary operator U ∈ B(K) such that 1 + zT 1 − zT = I * 1 + zU 1 − zU I for all z ∈ D, or equivalently T k = I * U k I for all positive integers k.According to the Nagy dilation theorem these operators T are precisely the contractions.A principal reason that the Nagy dilation theorem has had such an impact on operator theory was Sarason's discovery [19] of a geometric interpretation of the operator-theoretic notion of a dilation in terms of semi-invariant subspaces -see the Appendix for a brief account of the theory.In Sections 5-8 we amend Sarason's analysis to apply to equation (1.9) in the case δ > 0. We are led to the notion of an "even stranger dilation", where the unitary operator U in equation (1.9) is replaced by a contraction Y acting on a finite-dimensional space.
Definition 1.11.Let T ∈ B(H) where H is a finite-dimensional Hilbert space and assume that σ(T ) ⊆ K δ .We say that a triple (L, E, Y ) is an even stranger dilation of T relative to K δ if L is a Hilbert space with dim L = 2 dim H, E : H → L is an isometry, Y ∈ B(L) is a contraction, and (1.12) In Section 5 we prove the following proposition (Proposition 5.3 below).
Proposition 1.13.Let δ ∈ (0, 1), let T ∈ B(H), let dim H be finite.Assume that T is generic for K δ .Then W (T ) ⊆ K δ if and only if T has an even stranger dilation relative to K δ .
The first dividend of this proposition and Proposition 7.7 below is the following characterization of operators T such that W (T ) ⊆ K δ .
Theorem 1.14.Let δ ∈ (0, 1).Let dim H be finite and let T ∈ B(H) be generic for K δ .Then the following conditions are equivalent.
(ii) For some (equivalently for all) choice of Q commuting with T and satisfying Q 2 = T 2 − 4δ, there exists an invertible S ∈ B(H) such that 1 2 Theorem 1.14 is central to subsequent results of the paper.
1.4.Germinators and the geometric structure of even stranger dilations Theorem 1.14 enables us to prove in Theorem 8.25 that, for generic T with W (T ) ⊆ K δ , T can be represented in the form for some operator C which is a (δ, H)-germinator, in the sense of the following definition.
Definition 1.17.Let δ ∈ (0, 1).If H is a finite-dimensional Hilbert space and dim H = n, we say that C is a (δ, H)-germinator if C ∈ B(H ⊕ H) and C satisfies the conditions: C has 2n distinct eigenvalues, ( where C 12 is the 1-2 entry of C when C is represented as a 2 × 2 block matrix acting on H ⊕ H. 1.5.An Ando theorem for the ellipse In Section 9 we exploit (δ, H)-germinators, as defined in the previous subsection, to prove a generalization of Ando's theorem to the operators with numerical range in K δ .
Theorem 1.22.(Ando's theorem) For any bounded linear operator T on a Hilbert space, the numerical range W (T ) ⊆ D − if and only if T admits a factorization for some self-adjoint contraction A and some contraction B.
Theorem 9.18 implies the following result.
Theorem 1.23.Let H be a finite dimensional Hilbert space, let T ∈ B(H) and let δ ∈ [0, 1).W (T ) ⊆ K δ if and only if there exist a pair of contractions A, B ∈ B(H) such that A is self-adjoint and

A connection to Douglas-Paulsen operators
In Section 10 we probe yet another application of Theorem 1.14 and of (δ, H)-germinators.For δ ∈ [0, 1) we adopt the notation (1.24) For any δ ∈ (0, 1) an operator T on some Hilbert space H is said to be a Douglas-Paulsen operator with parameter δ if T ≤ 1 and T −1 ≤ 1/δ.The name recognizes work of R. G. Douglas and V. Paulsen [11] which gave a dilation theory for Douglas-Paulsen operators.An important step in their theory was the following estimate.If X is a Douglas-Paulsen operator with parameter δ, σ(X) ⊆ R δ and ϕ is a bounded holomorphic matrix-valued function on R δ then This result allowed them to show that if T ∈ B(H) is a Douglas-Paulsen operator, then there exists an invertible S ∈ B(H) such that and ST S −1 dilates to a normal operator with spectrum contained in the boundary ∂R δ .
In the scalar case a slightly stronger result had been obtained earlier by A. Shields [20, Proposition 23], with the smaller constant 2 + 1+δ 1−δ on the right hand side.He asked whether the constant 2 + 1+δ 1−δ could be replaced by a quantity that remains bounded as δ → 1.This question was answered in the affirmative by C. Badea, B. Beckermann and M. Crouzeix [5] and subsequently a better constant was established by M. Crouzeix [8].
It is relatively straightforward to show that if X is a Douglas-Paulsen operator with parameter δ then W (X + δX −1 ) ⊆ K δ .However, not every operator with numerical range in K δ is itself of the form X + δX −1 for some Douglas-Paulsen operator X with parameter δ (see Fact 10.3).It is, though, the case that every operator T such that W (T ) ⊆ K δ can be extended to a Douglas-Paulsen operator with parameter δ, as is summarized in the following statement, which follows from Theorem 10.6 in the body of the paper.Theorem 1.27.Let H be a finite-dimensional Hilbert space, let T ∈ B(H) and let δ ∈ (0, 1).W (T ) ⊆ K δ if and only if there exists a Hilbert space K containing H and a Douglas-Paulsen operator X ∈ B(K) with parameter δ such that H is invariant under X + δX −1 and T is the restriction of X + δX −1 to H.

The B. and F. Delyon family for an ellipse
For δ ∈ [0, 1) we adopt the notation (1.28) In Section 11 we shall describe relations between holomorphic functions on the elliptical region 2 G δ and on the annulus R δ that arise from the map π : R δ → G δ , π(z) = z + δ/z, which maps R δ onto G δ in a 2-to-1 manner.
2 G δ is the interior of K δ introduced previously.
We recall from [2, Chapter 9] that the B. and F. Delyon family F bfd (C) corresponding to a bounded open convex set C in C is the class of operators T such that the closure of the numerical range of T , W (T ) ⊆ C. The nomenclature is a tribute to the ground-breaking theorem of the brothers B. and F. Delyon [9], which asserts that, for any T ∈ B(H) such that W (T ) ⊆ C and any polynomial p, p(T ) ≤ κ(C) sup defined for ϕ ∈ Hol(R δ ), and defined for ϕ ∈ Hol(G δ ).There is no guarantee that the quantities defined by equations (1.29) and (1.30) are finite.Accordingly, we introduce the associated Banach algebras The relationship described in Theorem 1.27 between the B. and F. Delyon class corresponding to the elliptical region G δ and the Douglas-Paulsen operators with parameter δ induces the following inequality Surprisingly, Theorem 11.25 from Section 11 below shows that the reverse inequality also holds, so that ϕ bfd = ϕ • π dp for all ϕ ∈ Hol(G δ ).
We shall say that a function f ∈ Hol(R δ ) is symmetric with respect to the involution λ → δ/λ of R δ if f (λ) = f (δ/λ) for all λ ∈ R δ .By noting that a holomorphic function f on R δ has the form ϕ • π for some ϕ ∈ Hol(R δ ) if and only if f is symmetric with respect to the involution z → δ/z on R δ , we obtain a precise relation between the spaces H ∞ dp (R δ ) and H ∞ bfd (G δ ) (see Theorem 11.25 below).
1.8.An extremal problem for analytic functions on the symmetrized bidisc In the previous subsection we described results of Section 11 which show how the B. and F. Delyon norm • bfd can be interpreted in terms of function theory on R δ .In Section 12 we show that • bfd on Hol(G δ ) can also be interpreted in terms of function theory on the symmetrized bidisc G, the domain in C 2 defined by We observe that, for δ ∈ (0, 1) and s ∈ C, a point (s, δ) ∈ C 2 belongs to G if and only if s ∈ G δ .Thus, if we are given a function ϕ ∈ Hol(G δ ), we could look to express ϕ as the restriction to G δ of a function Φ ∈ Hol(G).Oka's Extension Theorem tells us there does exist Φ ∈ Hol(G) such that Φ(s, δ) = ϕ(s) for all s ∈ G δ .We ask further, for which ϕ ∈ Hol(G δ ) can Φ be chosen bounded, and what is the minimal H ∞ norm of all Φ that extend ϕ?
The next theorem gives a full answer to these questions.

A condition for W (T ) to be in an ellipse
Let δ ∈ [0, 1).We have introduced the open and closed elliptical sets G δ and K δ defined in equations (1.28) and (1.1).Their common boundary is We shall make extensive use of the Zhukovskii map, which relates an annulus to the elliptical region G δ .It is widely used in engineering applications, especially in aerodynamics, in the design of aerofoils [13, page 677].
The observations in the preceding paragraph imply that for each θ, the set H θ of complex numbers defined by is the supporting half plane to K δ at f (e iθ ).Since W (T ) ⊆ K δ if and only if W (T ) ⊆ H θ for all θ, and W (T ) ⊆ H θ if and only if Re e iθ f ′ (e iθ )(T − f (e iθ )) ≥ 0, it follows that If X is an invertible operator, then

Strange dilation on an ellipse
In this section we shall combine the Herglotz Representation Theorem for operatorvalued functions with Proposition 2.2 to obtain an analog of Berger's Theorem for K δ .
We first recall the following generalization of the Herglotz Representation Theorem due to Naimark [17,16].Theorem 3.1.Let H be a Hilbert space and assume that V is an analytic B(H)-valued function defined on D satisfying Re V (z) ≥ 0 for all z ∈ D. Then there exist a Hilbert space K, an isometry I : H → K, and a unitary operator U ∈ B(K) such that We shall massage G to produce a closely related function F , which has the virtue that it gives rise to a dilation theorem for operators with numerical range in an elliptical region that is closely analogous to Berger's strange dilation theorem (see Theorem 1.6 above).
We shall now prove Theorem 1.8 from the Introduction.
Theorem 3.5.Let δ ∈ [0, 1).Let T ∈ B(H) be an operator satisfying σ(T ) ⊆ K δ , and let F : D → B(H) be defined by the formula Then W (T ) ⊆ K δ if and only if there exists a Hilbert space K ⊇ H, an isometry I : H → K, and a unitary operator U ∈ B(K) such that Proof.Proposition 2.2 and Theorem 3.1 imply that W (T ) ⊆ K δ if and only if, for the analytic function G : D → B(H) defined by there exists a Hilbert space K, an isometry I : H → K, and a unitary operator U ∈ B(K) such that Note that, for all z ∈ D, Hence, for all z ∈ D, By Lemma 3.2, for all z ∈ D, Thus if and only if Therefore, W (T ) ⊆ K δ if and only if there exists a Hilbert space K ⊇ H, an isometry I : H → K, and a unitary operator U ∈ B(K) such that In light of the obvious similarity of Theorem 1.6 and Theorem 3.5, and in honor of Berger's seminal contribution [6] we make the following definition.Definition 3.8.Let δ ∈ [0, 1), and let T ∈ B(H) be an operator satisfying σ(T ) ⊆ K δ .We say that a triple (K, I, U) is a strange dilation of T relative to K δ if K is a Hilbert space, I : H → K is an isometry, U ∈ B(K) is unitary, and the formula Where the role of K δ is clear from the context, we will omit "relative to K δ " when speaking of strange dilations.Proposition 3.10.Let δ ∈ (0, 1), let µ ∈ G δ , and let T be the scalar operator of multiplication by µ on a (finite-or infinite-dimensional) Hilbert space H.Then, for any strange dilation (K, I, U) of T , K has infinite dimension.
Proof.Assume that T has the strange dilation (K, I, U), so that To see that K must be infinite-dimensional, suppose to the contray that dim K = n < ∞.
Then there is an orthonormal basis e 1 , . . ., e n of K consisting of eigenvectors of U, with corresponding eigenvalues τ 1 , . . ., τ n , all of which have unit modulus.Let x be a nonzero vector in H.We have Write x i = Ix, e i K for i = 1, . . ., n, so that Ix = n i=1 x i e i .Observe that since x = 0 and I is an isometry, the components x 1 , . . ., x n of Ix are not all zero.Choose j ∈ {1, . . ., n} such that x j = 0 and notice that I * e j = 0, since 0 = x j = x, I * e j .We have By Lemma 3.2 the rational function 1−µz+δz 2 is analytic on D. One can check that, when µ ∈ G δ , the quadratic 1 − µz + δz 2 has no zeros on the unit circle T. Thus F (z) is analytic on D − .As z → τj in equation (3.12) the left hand side tends to a limit in H, while the right hand side is unbounded.This contradiction shows that K is not finite-dimensional.

Prepairs for generic operators T and residues
Let δ ∈ (0, 1), and let If T ∈ B(H) is generic for K δ and n = dim H, then there exist n distinct points µ 1 , . . ., µ n ∈ C (the eigenvalues of T ) and n vectors e 1 , . . ., e n ∈ H (the corresponding eigenvectors of T ) such that Once such an enumeration of the eigenvalues of T is chosen, then T is generic precisely when the equation π(λ) = µ i has two distinct solutions for each i.If for each i, we let λ 1 i , λ 2 i be an enumeration of these two solutions, then we may define a pair of operators X 1 , X 2 ∈ B(H) by the formulas X r e i = λ r i e i for r = 1, 2 and i = 1, . . ., n.
Evidently, if X 1 and X 2 are so defined, then Definition 4.7.Let δ ∈ (0, 1).Let T ∈ B(H) be generic for K δ and let dim H = n.We say that a pair X = (X 1 , X 2 ) where We say that X is a generic prepair for T if, in addition, both X 1 and X 2 have n distinct eigenvalues.
Remark 4.8.Let T ∈ B(H) be generic for K δ and let X = (X 1 , X 2 ) be a generic prepair for T .It is easy to see that Lemma 4.10.Let δ ∈ (0, 1), let T ∈ B(H) and let σ(T ) ⊆ K δ .Then T is generic for K δ if and only if there exists a generic prepair for T .
Proof.If T is generic for K δ then the construction described in the paragraph following Definition 4.1 shows that there is a generic prepair for T .Conversely, suppose that T ∈ B(H) and σ(T ) ⊆ K δ , and suppose that there exists a generic prepair X = (X 1 , X 2 ) for T , where X 1 , X 2 ∈ B(H).By property (iii) of Definition 4.7, X 1 X 2 = δ, and so X 1 and X 2 are both nonsingular since dim H = n < ∞.Therefore 0 does not lie in either σ(X 1 ) or σ(X 2 ).By assumption X = (X 1 , X 2 ) is a generic prepair for T , and so, for j = 1, 2, X j has n distinct eigenvalues λ j 1 , . . ., λ j n .Let corresponding eigenvectors of X 1 be e 1 , . . ., e n .Since also X 1 X 2 = δ, we have X 2 = δ(X 1 ) −1 , and so (possibly after renumbering) λ 2 j = δ(λ 1 j ) −1 for j = 1, . . ., n, and an eigenvector of X 2 corresponding to λ 2 j is the eigenvector e j of X 1 corresponding to λ 1 j .Further, T = X 1 + X 2 = X 1 + δ(X 1 ) −1 , and so e j is an eigenvector of T corresponding to the eigenvalue We denote by P the set of polynomials over C. Note that if T ∈ B(H) has a generic prepair then T has exactly 2 n generic prepairs.If T is generic for K δ , then for each choice of generic prepair X = (X 1 , X 2 ) one can define a pair of residues A = (A 1 , A 2 ) to be the unique pair of operators in the algebra generated by T satisfying It is easy to verify that and A 2 = 1 2 for all choices of generic prepairs.The following lemma gives a valuable formula which relates prepairs and residues to strange dilations.Lemma 4.12.Let δ ∈ (0, 1).Suppose that T ∈ B(H) is generic for K δ and X is a generic prepair for T with residues A. If (K, I, U) is a strange dilation of T , then for all p ∈ P.
Proof.Under the assumptions of the lemma, since (K, I, U) is a strange dilation of T , then the formula holds for all z ∈ D. The formula (4.11) implies that, for all z ∈ D, If we expand the analytic functions in this formula into power series and equate coefficients we obtain the relations The lemma follows if one takes linear combinations of equations of the form (4.14).

Cutting down strange dilations
We have shown in Proposition 3.10 that if T is the operator of multiplication by a scalar µ ∈ G δ on a finite-or infinite-dimensional Hilbert space H, then necessarily, for any strange dilation (K, I, U) of T , K is infinite dimensional.One can obtain a finitedimensional dilation of T if one is willing to replace the unitary U in Definition 3.8 with a contraction Y .This idea is formalized in the following definition.Definition 5.1.Let δ ∈ (0, 1) and let T ∈ B(H) where H is a finite-dimensional Hilbert space and assume that σ(T ) ⊆ K δ .We say that a triple (L, E, Y ) is an even stranger dilation of T relative to K δ if L is a Hilbert space with dim L = 2 dim H, E : H → L is an isometry, Y ∈ B(L) is a contraction, and We may omit the phrase "relative to K δ " when it is clear from the context.
Proposition 5.3.Let δ ∈ (0, 1), let T ∈ B(H) and let dim H be finite.Assume that T is generic for K δ .Then T has a strange dilation relative to K δ if and only if T has an even stranger dilation relative to K δ .
Proof.First assume that T has a strange dilation (K, I, U) relative to K δ , as in Definition 3.8.Fix a generic prepair X = (X 1 , X 2 ) of T with residue A = (A 1 , A 2 ) and assume that e i is an eigenvector of T corresponding to the eigenvalue µ i , as in Definition 4.1 and λ r i is the corresponding eigenvalue of X r , so that X r e i = λ r i e i for r = 1, 2 and i = 1, . . ., n. ( For i = 1, . . ., n, let I i = {p ∈ P : p(λ r i ) = 0, r = 1, 2} and define closed linear subspaces M i and N i in K by Since for each i, both M i and N i are invariant for U, so also M and N are invariant for U. Consequently, if we define a Hilbert space L by for all p ∈ P. Also, since for each i, Therefore, ran I ⊆ M. Also observe that if p ∈ I i and y = p(U)Ie i , then by Lemma 4.12 ))e i = p(λ 1 i )A 1 e i + p(λ 2 i )A 2 e i = 0. Therefore, N i ⊆ ker I * for each i.Consequently, N ⊆ ker I * , or equivalently, ran I ⊆ N ⊥ .As we have both ran I ⊆ M and ran I ⊆ N ⊥ , ran I ⊆ L. (5.7) Now observe that if we define E : H → L, by the formula then inclusion (5.7) implies that E is an isometry.Also, using formula (5.5), we deduce that if k ≥ 0, then that is, the formula (5.2) holds for all z ∈ D.
To summarize, we have shown that E is an isometry and the formula (5.2) holds for all z ∈ D. It is clear that Y = P L U|L is a contraction, since U is unitary.
There remains to show that dim L = 2 dim H.For each i = 1, . . ., n and r = 1, 2, choose χ r i ∈ P such that for each i and r.
Note that if p ∈ P, then, by equation (5.8) and by Lemma 4.12, where δ k r is the Kronecker symbol.We claim that the set S of vectors in L defined by is linearly independent.For if c r i are complex numbers with i,r c r i χ r i (Y )Ie i = 0, and p ∈ P, then by equation (5.9), But since p ∈ P is arbitrary and the residues A 1 and A 2 are invertible, this calculation shows that the set S is linearly independent.In particular, dim L ≥ 2n.Hence, in light of inequality (5.6), dim L = 2n, as was to be proved.Now assume that (L, E, Y ) is an even stranger dilation of T relative to K δ .As in particular, Y is a contraction, by the Nagy Dilation Theorem, there exists a Hilbert Space K, an isometry F : L → K, and a unitary U ∈ B(K) such that for all p ∈ P. Consequently, if we set I = F E, then it follows that (K, I, U) is a strange dilation of T relative to K δ .

The structure of even stranger dilations
In this section we characterize generic operators T with W (T ) ⊆ K δ using even stranger dilations of T .Lemma 6.1.Let δ ∈ (0, 1) and let T ∈ B(H) be generic for K Proof.If (K, E, Y ) is an even stranger dilation of T relative to K δ , then equations (4.11) and (5.2) imply that and S 2 are finite sets.Furthermore, the left hand side of equation (6.2) is analytic on C \ S 1 and the right hand side of equation (6.2) is analytic on C \ S 2 .Since 1−zX 2 and E * 1 1−zY E are analytic functions on C \ (S 1 ∪ S 2 ) that agree on a nonempty open set, they agree on C \ (S 1 ∪ S 2 ).Therefore, equation (6.2) holds for all z ∈ C \ (S 1 ∪ S 2 ).
The previous paragraph showed that σ(X 1 ) ⊆ σ(Y ).By a similar argument it follows that σ(X 2 ) ⊆ σ(Y ) as well.Since X = (X 1 , X 2 ) is a generic prepair, σ(X 1 ) ∪ σ(X 2 ) consists of 2 dim H points and dim . ., 2n.The vectors u i , i = 1, . . ., 2n, and v i , i = 1, . . ., 2n, comprise bases for H ⊕ H and K respectively, since they are both 2n in number and correspond to distinct eigenvalues.Define an operator It follows that S ∈ B(H ⊕ H, K) is an invertible operator such that Y = S(X 1 ⊕ X 2 )S −1 .Lemma 6.3.Let δ ∈ (0, 1) and let T ∈ B(H) be generic for K δ .Assume that X = (X 1 , X 2 ) is a generic prepair for T .Then (K, E, Y ) is an even stranger dilation of T relative to K δ if and only if there exist an invertible S ∈ B(H⊕H, K) and C ∈ B(H, H⊕H) such that Y = S(X 1 ⊕ X 2 )S −1 and E = SC, (6.4) and where S and C satisfy the following three conditions.
then, for each i = 1, 2, C i is invertible and commutes with X i and T .(iii) If C is decomposed as in Condition (ii) above, then Proof.First assume that (K, E, Y ) is an even stranger dilation of T .The formula holds for all z ∈ D. The formula (4.11) implies that, for all z ∈ D, If we expand the analytic functions in this formula into power series and equate coefficients we obtain Let us take linear combinations of the last equations to get for all p ∈ P. By Lemma 6.1, there exists an invertible S ∈ B(H ⊕ H, K) such that then, by equations (6.6), (6.7) and the definitions of B and C, it follows that for all p ∈ P, By assumption, X is a generic prepair for T , and so σ(X 1 ) and σ(X 2 ) are disjoint and finite.Hence there exists a polynomial p 0 such that p 0 (z) = 1 for all z ∈ σ(X 1 ) and p 0 (z) = 0 for all z ∈ σ(X 2 ).In equation (6.8) replace p by pp 0 and observe that p 0 (X 1 ) = 1 and p 0 (X 2 ) = 0, and so equation (6.8) becomes for all p ∈ P. Similarly, for all p ∈ P. Let p = 1 in equations (6.9) and (6.10), to get which implies that C 1 and C 2 are invertible.Let p = z in equations (6.9) and (6.10), to get 1 2 which implies that C 1 commutes with X 1 and C 2 commutes with X 2 .By equation (4.9), Therefore, C 1 and C 2 commute with T .Finally, to see that Condition (iii) holds, let B be as in the previous paragraph and compute that and that S and C satisfy Conditions (i), (ii), and (iii).We wish to show that if Y and E are defined by formulae (6.4), then (K, E, Y ) is an even stranger dilation of T .First observe that since There remains to show that E is an isometry and that equation (5.2) holds.
Using Condition (iii) we see that Therefore E is an isometry.
To prove that equation (5.2) holds, fix z ∈ D and note that by equation (4.11).
The proof of Lemma 6.3 is complete.
7. The existence of even stranger dilations Proposition 7.1.Let δ ∈ (0, 1) and let T ∈ B(H) be generic for K δ .Assume that X = (X 1 , X 2 ) is a generic prepair for T .There exists an even stranger dilation of T relative to K δ if and only if there exists a self-adjoint operator ∆ ∈ B(H ⊕ H) satisfying and Furthermore, if ∆ is a self-adjoint operator in B(H ⊕ H) satisfying conditions (7.2) and (7.3), then ∆ is strictly positive definite.
Proof.First assume that (K, E, Y ) is an even stranger dilation for T .Let S and C be as in Lemma 6.3 and define ∆ by We have by equation (6.11) by Condition (i) of Lemma 6.3 = ∆, that is, inequality (7.2) holds.Also, that is, equality (7.3) holds.Now assume that ∆ is a self-adjoint operator in B(H ⊕ H) satisfying the relations (7.2) and (7.3).We first show that ∆ is strictly positive definite and then show that there exists an even stranger dilation for T .
To construct an even stranger dilation for T , let K = H ⊕ H, define E by and let Then dim K = 2 dim H and the inequality (7.2) implies that Y ≤ 1.Also, as E * E = 1, E is an isometry.Finally, noting that equation (7.3) implies that ∆ we have, for z ∈ D, Observe that the above calculation is valid for any choice of a prepair (X 1 , X 2 ) for T , and so the existence statement in Proposition 7.1 does not depend on the choice of a generic prepair for T .
The following proposition, which is essentially equivalent to Proposition 7.1, gives necessary and sufficient conditions for the existence of an even stranger dilation for T directly in terms of T rather than in terms of a prepair X for T .If T ∈ B(H) is generic for K δ , then we may define Z T ∈ B(H ⊕ H) by where Q is any operator in B(H) commuting with T and satisfying Just as T has 2 dim H prepairs, there are 2 dim H choices for Q satisfying equation (7.6).Also, the notation 'Z T ' for the operator defined by formula (7.5) is slightly ambiguous in that it does not reflect the dependence of Z T on the choice of Q.
Proposition 7.7.Let δ ∈ (0, 1).If T ∈ B(H) is generic for K δ , then the following conditions are equivalent.(i) There exists an even stranger dilation of T relative to K δ .
(ii) For some (equivalently for all) choice of Q satisfying equation (7.6), there exists a strictly positive definite Γ ∈ B(H) such that (iii) For some (equivalently for all) choice of Q commuting with T and satisfying Q 2 = T 2 − 4δ, there exists an invertible S ∈ B(H) such that Proof.Suppose Condition (i) holds.By Proposition 7.1, for a generic for K δ T and a generic prepair X = (X 1 , X 2 ) for T , there exists an even stranger dilation of T if and only if there exists a self-adjoint operator ∆ ∈ B(H ⊕ H) satisfying If we define a symmetry U ∈ B(H ⊕ H) by where Q = X 1 − X 2 is an operator in B(H).By the definition 4.7 of a prepair X = (X 1 , X 2 ) for T , the equations X Hence, the inequality (7.10) is equivalent to Z * T U∆U Z T ≤ U∆U.Also, equation (7.11) is equivalent to Therefore, if we assume that ∆ satisfies the relations (7.10) and (7.11), then inequality (7.8) holds if we let 1 0 0 Γ = 2U∆U.This proves that Condition (i) implies Condition (ii).Suppose Condition (ii) is satisfied.That is, there is a strictly positive definite Γ ∈ B(H) such that equations (7.8) holds.Since Γ > 0, σ(Γ) is a compact subset of (0, ∞), and so 0 / ∈ σ(Γ).Thus Γ is invertible and Condition (iii) is satisfied too.There remains to prove that Condition (iii) implies Condition (i).Let Condition (iii) hold, so that, for some choice of an operator Q in B(H) commuting with T and satisfying Q 2 = T 2 − 4δ, there exists an invertible S ∈ B(H) such that Then Since, by definition, we have One can see that, ∆ defined by equations 7.15 and 7.14, is a self-adjoint operator ∆ ∈ B(H ⊕ H).
By assumption of Condition (iii), S ∈ B(H) is invertible and such that In view of equation (7.16), the above inequality is equivalent to and so U∆U − Z * T U∆U Z T ≥ 0. We have shown at the beginning of the proof that the last inequality implies that ∆ satisfies Let T ∈ B(H) be generic for K δ .Recall that Z T ∈ B(H ⊕ H) is defined by where Q is any operator in B(H) commuting with T and satisfying Proof.Since T is generic for K δ , σ(T ) ⊆ K δ .By Theorem 3.5, W (T ) ⊆ K δ if and only if there is a strange dilation of T .By Proposition 5.3, for T which is generic for K δ , T has a strange dilation if and only if T has an even stranger dilation.By Proposition 7.7, for T which is generic for K δ , T has an even stranger dilation if and only if the following condition is satisfied: for some (equivalently for all) choice of Q commuting with T and satisfying Q 2 = T 2 − 4δ, there exists an invertible S ∈ B(H) such that The statement of Theorem 8.3 follows.
This theorem suggests that there should be a parametrization of the operators T ∈ B(H) for which W (T ) ⊆ K δ in terms of contractions C that act on H ⊕ H.In the remainder of this section we shall explicitly describe such a parametrization.
When H is a Hilbert space we let J denote the symmetry in B(H ⊕ H) defined by If T ∈ B(H) is generic for K δ and W (T ) ⊆ K δ , then using Theorem 8.3 we may define C ∈ B(H ⊕ H) by the formula Evidently, then C is a contraction, that is, Also, let dim H = n and σ(T ) = {µ 1 , . . ., µ n }.As we showed in Proposition 7.7, since T is generic for K δ and W (T ) ⊆ K δ , T has an even stranger dilation, and hence the implication (i) =⇒ (ii) of Proposition 7.7 shows that, for every choice (X 1 , X 2 ) of generic prepair for T , U(X 1 ⊕ X 2 )U = Z T , where (see equation (7.12)).Note that U * = U, so that Z T is unitarily equivalent to X 1 ⊕ X 2 , and therefore As a consequence, as T is generic for K δ , C is generic, that is, To prove equation (8.9) first observe that The last equation follows from the facts that QT = T Q and Q 2 = T 2 − 4δ.Therefore, A final property of C that we wish to observe is that 0 ∈ σ(C Armed with this definition, we summarize the above observations in the following proposition. Proposition 8.16.Assume that T ∈ W δ (H) is generic for K δ .There exists an invertible S ∈ B(H) such that if C is defined by the equation then C is a (δ, H)-germinator.
A converse to Proposition 8.16 hinges on the inversion of equation (8.6) which we turn to now.Lemma 8.18.If T ∈ B(H) is generic for K δ , then Proof.By equation (8.9), Z T is J-equivalent to δZ −1 T .Thus where C 2 is similar to C 1 .
Proof.Since C is a (δ, H)-germinator, δ C = JCJ, and so we see from the 1-2 entry that Therefore, the relation (8.15) 0 ∈ σ(C 12 ), implies that C 12 is invertible and Proposition 8.22.Let H be a finite-dimensional Hilbert space and dim H = n, and let C ∈ B(H ⊕ H) be a (δ, H)-germinator, then there exists an invertible S ∈ B(H) and a T ∈ W δ (H) such that T is generic for K δ and the equation Proof.First observe using equation (8.14) that if Thus, using the fact (8.13) that C has 2n distinct eigenvalues, we may decompose the set of eigenvalues of C into a disjoint union Since all eigenvalues of C lie in R δ , σ(C) ⊆ R δ .Furthermore, if u i v i and u i −v i are the eigenvectors corresponding to λ i and δ/λ i respectively, then the statement (8.13), that C has 2n distinct eigenvalues, implies that {u 1 , u 2 , . . ., u n } is a basis for H. Therefore, there exist S, T ∈ B(H) satisfying and the operator Q defined by is a square root of T 2 − 4δ that commutes with T .Observe that T is the restriction of C + δ C to H ⊕ {0} which is the span of u 1 , . . ., u n , and so By the definition of T , the eigenvalues of T are the points Therefore the equation π(λ) = µ i has two distinct solutions in R δ for each i.Hence T is generic for K δ .
To see that the equation ( 8.23) holds it suffices to show that the two operators agree on the eigenvectors of C (which the relation (8.13) guarantees to be a basis for H ⊕ H).But for fixed i, and likewise, then C is a (δ, H)-germinator.
Conversely, suppose that C ∈ B(H⊕H) is a (δ, H)-germinator and there is an invertible operator S ∈ B(H) such that By the definition of a (δ, H)-germinator, C ≤ 1, and so equation is satisfied.Thus, by Theorem 8.3, T ∈ W δ (H).
Note that, by equation (8.26), Therefore, 9. A parametrization of the operators T with W (T ) ⊆ K δ In view of Theorem 8.25 it is relatively straightforward to parametrize W δ (H).
Lemma 9.1.Let δ ∈ (0, 1), let H be a Hilbert space of finite dimension n and let C ∈ B(H ⊕ H).Suppose that the operator C satisfies the relations Then C = √ δXJ for some X having the block matrix representation with respect to the decomposition and where the operator Proof.Suppose that C satisfies the relations (9.2).Then δ = CJCJ.
x for all x ∈ H ⊕ H, it follows that ker(1−X)+ker(1+X) = H⊕H.Since clearly ker(1−X)∩ker(1+X) = {0}, H ⊕ H is the vector space direct sum of ker(1 − X) and ker(1 + X).Let us write for the eigenspaces of X corresponding to the eigenvalues 1, −1 respectively.The operator matrix of X relative to the orthogonal decomposition we have also X 2 22 = 1 and therefore σ(X 22 ) ⊆ {1, −1}, by the spectral mapping theorem.However, V ⊥ − does not contain any eigenvector of X corresponding to the eigenvalue 1, and so σ(X 22 ) = {−1}.Thus 1 − X 22 is invertible, and so The expression of X by an operator matrix with respect to the direct decomposition , and so δX * X ≤ 1.Thus For general matrices A, B, D of appropriate sizes the identity and Proof.Using Lemma 9.1 we see that where X is as in the relations (9.3), (9.5), and ⊥ we obtain the formula A is a self-adjoint contraction, as required by the statement of the lemma.Furthermore, where U 1 and U 2 are Hilbert space isomorphisms.In terms of these new operators, equation (9.10) becomes Finally, if we define F = −U * 1 EU 2 , equation (9.11) becomes equation (9.9) and as U 1 and U 2 are Hilbert space isomorphisms, equation (9.5) implies equation (9.8).Lemma 9.12.Let δ ∈ (0, 1) and let H be a finite dimensional Hilbert space.If T ∈ W δ (H) is generic for K δ , then there exist a pair of contractions A, Y ∈ B(H) such that A is self-adjoint and Proof.Assume that T ∈ W δ (H) is generic for K δ .By Theorem 8.25, there exists a (δ, H)-germinator C such that (8.26) holds.In particular, By Lemma 9.7, there exist A, F ∈ B(H) such that A is a self-adjoint contraction such that (9.8) and (9.9) hold.If we define then F ≤ 1 and the equation (9.13) holds.
For c ∈ C and r ≥ 0 we let D c (r) denote the closed disc in the complex plane centered at c of radius r, that is, The following lemma describes the radii of the maximal discs in K δ centered on the major axis of K δ .Lemma 9.14.Let δ ∈ (0, 1).If, for t ∈ [−(1 + δ), 1 + δ], we define ], the maximal disc always touches the boundary of K δ in two distinct points symmetric about the x-axis and the formula for ρ(t) can be justified by Lagrange multipliers.For t ∈ [− 4δ 1+δ , 4δ 1+δ ], the maximal disc touches the boundary of K δ in a single point ±(1 + δ), which explains the formula for ρ(t) in that case.
It was slightly surprising to the authors that the role played by the transition points ± 4δ 1+δ in the preceding lemma was not played by the foci (±2 √ δ).However, the foci of K δ appear in the following corollary to Lemma 9.14.
By passing to a subsequence if necessary we may assume that there exist A, Y ∈ B(H) such that This is a true statement, by Ando's Theorem [4, Theorem 1].We can therefore consider Theorem 9.18 as a generalization of Ando's theorem to operators T with W (T ) ⊆ K δ in the finite-dimensional case.
10. Operators T with W (T ) ⊆ K δ and Douglas-Paulsen operators In this section we present yet another consequence of Theorem 8.25, a tight connection between the operators with numerical range in an ellipse and Douglas-Paulsen operators.Consider a number δ ∈ (0, 1).In [11] R. G. Douglas and V. Paulsen developed a dilation theory for operators T on some Hilbert space H such that δ 2 ≤ T * T ≤ 1. Accordingly, we shall call any such operator a Douglas-Paulsen operator with parameter δ.
More exposition and references on such operators can be found in [2, Chapter 9].
Proof.Let H be n-dimensional and notice that σ(X) ⊆ R − δ .Set T = π(X) = X + δ/X.We first prove the proposition under the assumptions that and both X and δ/X have n distinct eigenvalues.The hypotheses imply that (X, δ/X) is a generic prepair for T .We observe that the fact ensures (with the aid of Lemma 4.10) that T is generic for K δ .Furthermore, as and X − δ/X is an operator that commutes with T and satisfies (X − δ/X) 2 = T 2 − 4δ, we may choose Q = X − δ/X in the definition of Z T , equation (7.5), and we have where Consequently, as X , δ/X ≤ 1 and U is unitary, Z T ≤ 1. Hence condition (8.4) holds (with S = 1) and Theorem 8.3 implies that W (T ) ⊆ K δ .
To relax the assumptions (10.2) and the assumption that X has n distinct eigenvalues, assume that X is now a general Douglas-Paulsen operator with parameter δ.Choose a sequence {X k } such that X k is a Douglas-Paulsen operator with parameter δ, X k has n distinct eigenvalues, the stated assumptions hold for all k and X k → X as k → ∞ for the weak operator topology.By the special case that has been proved above, W (π(X k )) ⊆ K δ for all k.Hence W (π(X)) ⊆ K δ .Proposition 10.1 asserts that π : X → X + δ/X maps the class of Douglas-Paulsen operators with parameter δ on a Hilbert space H into the class of operators on H whose numerical ranges are contained in K δ .
Fact 10.3.The map π is not onto, even in the case of a 2-dimensional H.

Proof. Consider the operator
]) shows that the numerical range of T is K δ , and so W (T ) ⊆ K δ .Note that T has eigenvalues .
We note that these two quantities in equations (11.1) and (11.2) may be infinite.The results of Section 10, which describe an intimate connection between the families F dp (δ) and F bfd (G δ ), give rise to a relationship between the norms • dp and • bfd , and between the associated Banach algebras This relationship is formalised in Theorem 11.25 below.The proof of Theorem 11.25 will require a number of ingredients.The first one is contained in Propositions 11.9 and 11.16 which show that, instead of taking the supremum in the foregoing formulae, equations (11.1) and (11.2), over all operators in F dp (δ) and F bfd (G δ ) respectively, it is sufficient to take the supremum only over the operators in the corresponding families that act on finite-dimensional Hilbert spaces.
A second ingredient that we shall require for the proof of Proposition 11.16 is the notion of the Badea-Beckermann-Crouzeix family corresponding to a pair (c, r), where, for some positive integer m, c = (c 1 , . . ., c m ) and r = (r 1 , . . ., r m ), each c j ∈ C and each r j > 0. Such a pair (c, r) is called (in [2, Section 9.6]) a bbc pair, and corresponding to such a pair the bbc family F bbc (c, r) is defined to be the family of operators T such that the spectrum σ(T ) is contained in the set A final ingredient is a useful result from [1], stated below as Theorem 11.6, which sheds light on the dp and bbc norms -see Proposition 11.9 below.For the convenience of the reader we recall from [1] that, corresponding to a domain of holomorphy U in C d and an m-tuple γ = (γ 1 , . . ., γ m ) of holomorphic functions on U we denote5 by E γ the domain and by F γ the family of commuting d-tuples T = (T 1 , . . ., T d ) of Hilbert space operators such that σ(T ) ⊆ E γ and γ j (T ) ≤ 1 for j = 1, . . ., m, where σ(T ) denotes the Taylor spectrum of the d-tuple T of commuting operators.Using F γ one defines a norm • γ on Hol(E γ ) thus: for any ϕ ∈ Hol(E γ ), and then one defines H ∞ γ to be the set of functions ϕ ∈ Hol(E γ ) such that ϕ γ < ∞.Here, ϕ(T ) is defined by the Taylor functional calculus (though in this paper we only require the case that d = 1, so that the classical functional calculus for operators is sufficient).
Two examples of families expressible in the form F γ that we shall exploit are the following.The norm • γ is in general quite complex and difficult to analyse, especially in several variables, depending as it does on the Taylor spectrum and the Taylor functional calculus.To make the norm more amenable to analysis, the authors of [1] introduced a more elementary approach to the norm • γ which bypasses the use of the Taylor spectrum and calculus, making use of diagonalizable matrices rather than general operators on Hilbert space, thereby making the notions of spectrum and functional calculus much simpler.To T ∈ F γ in this inequality we find that ψ γ ≤ 1, and so ϕ γ / ϕ γ,gen ≤ 1.Thus the equation (11.8) holds.
(ii) As we have shown in Example 11.4, for γ = (γ 1 , . . ., γ m ) where γ j is the polynomial (z − c j )/r j for j = 1, . . ., m, F γ = F bbc (c, r).The rest of the proof follows from Corollary 11.7 and is similar to the case (i).
Our goal is now to prove Proposition 11.16 which shows the supremum in the definition of ϕ bfd , instead of being taken over all operators in the B. and F. Delyon family, can equivalently be taken only over the operators in the family that act on a finite-dimensional Hilbert space.Since the family F bfd (G δ ) is not of the form F γ for any γ, we cannot deduce the desired equality directly from Corollary 11.7.We therefore approximate the bfd norm by the bbc norm corresponding to a suitable bbc pair, and then use Proposition 11.9 to show that the required inequality holds.
The approximation result we need, which is [2, Lemma 9.76], is the following.
Consequently, inequality (11.18) will follow if we can show that sup To prove inequality (11.19) fix T ∈ F bfd (G δ ), and let s ∈ [0, 1).Let C = sG δ .As C − is a compact subset of G δ , there exists ε > 0 such that To summarize, in the previous two paragraphs we have shown that if T ∈ F bfd (G δ ), s ∈ [0, 1), and η > 0, then there exists a matrix M ∈ F bfd (G δ ) such that the inequality (11.23) holds.This proves the inequality (11.19) and completes the proof of Proposition 11.16.Now observe that if ϕ ∈ Hol(G δ ) then we may define π ♯ (ϕ) ∈ Hol(R δ ) by the formula for all λ ∈ R δ .
We record the following simple fact from complex analysis without proof.
The following theorem gives an intimate connection between the • dp and • bfd norms.

.29)
Consider any matrix T ∈ F bfd (G δ ).We have W (T ) ⊆ G δ ⊂ K δ , and so Proposition 10.4 applies that there exists X ∈ B(H ⊕ H) such that X ≤ 1 and X −1 ≤ 1/δ and T = (X + δX −1 )| H ⊕ {0}.Thus T is a restriction of π(X) to an invariant subspace N = H ⊕ {0} for π(X).Therefore, Take the supremum of both sides over all matrices T ∈ F bfd (G δ ) to infer that We claim now that the mapping π ♯ maps H ∞ bfd (G δ ) surjectively onto the set of functions in H ∞ dp (R δ ) that are symmetric with respect to the involution λ → δ/λ.For consider any function f ∈ H ∞ dp (R δ ) that is symmetric with respect to the involution λ → δ/λ, that is, such that f (δ/λ) = f (λ) for all λ ∈ R δ .By Lemma 11.24, there exists a function ϕ ∈ Hol(G δ ) such that π ♯ ϕ = f .By the inequality (11.30), ϕ bfd ≤ f dp < ∞.Hence ϕ ∈ H ∞ bfd (G δ ) and π ♯ ϕ = f .12. The Banach algebra H ∞ bfd (G δ ) and the symmetrized bidisc In this section we shall give an interpretation of the B. and F. Delyon norm on Hol(G δ ) in terms of the solution to a certain extremal problem in two complex variables.Let G denote the symmetrized bidisc, the open set in C 2 defined by either of the equivalent formulas G = ρ(D 2 ) where ρ : C 2 → C 2 , the symmetrization map, is defined by ).The following simple lemma demonstrates that G δ may be viewed as a slice of G.
Lemma 12.1.For δ ∈ (0, 1), Evidently, the lemma implies that if Φ ∈ Hol(G), then we may define ϕ ∈ Hol(G δ ) by the formula ϕ(µ) = Φ(µ, δ) for all µ ∈ G δ .(12.2) Lemma 12.1 prompts the question: if ϕ ∈ Hol(G δ ), does there exist a function Φ ∈ Hol(G) such that equation (12.2) holds?Or, in other words.such that ϕ is the restriction of Φ to G δ ?A classical result relevant to this question is Cartan's Extension Theorem [7], which states that if V is an analytic variety in a domain of holomorphy U and ϕ is a holomorphic function on V then there is a holomorphic function Φ on U such that the restriction of Φ to V is ϕ.
The symmetrized bidisc G is a domain of holomorphy in C 2 and G δ is a variety in G, so Cartan's theorem applies to prove that there exists a function Φ ∈ Hol(G) such that ϕ is the restriction of Φ to G δ ; however, Cartan's theorem gives no information about bounds on Φ, even when ϕ is bounded on G δ .It is therefore natural to pose the following extremal problems: The next two propositions combine to show that the minimum m ϕ is ϕ bfd , so that m ϕ is finite if and only so that ψ = π ♯ (ϕ).Hence, using Theorem 11.25, we deduce that To see that Φ ∞ = ϕ bfd observe that the relations (12.9) and (12.11) imply that Φ ∞ ≤ ϕ bfd .As equation (12.2) holds, the reverse inequality, Φ ∞ ≥ ϕ bfd , follows from Proposition 12.4.Propositions 12.5 and 12.4 combine to yield the following statement.Theorem 12.13.Let δ ∈ (0, 1).For any ϕ ∈ Hol(G δ ), the minimum of Φ H ∞ (G) over all functions Φ ∈ H ∞ (G) that extend the function (s, δ) → ϕ(s) is ϕ bfd .In particular, an analytic function ϕ on G δ has a bounded extension Φ to G if and only if ϕ ∈ H ∞ bfd (G δ ).

Appendix on the numerical range and dilations
In this appendix we recall some known results on the numerical range W (T ) of an operator T and in particular the relationship between the inclusion of W (T ) in a prescribed disc and the existence of a dilation of T of a particular type.
The notion of a dilation of an operator has proved fruitful in the study of operators.It is a crucial element of the "harmonic analysis of operators on Hilbert space", the theory established by B. Sz.-Nagy and C. Foias in [14].Consider an operator T on a Hilbert space H, and let H be decomposed into an orthogonal direct sum of three subspaces H 1 , H 2 , H 3 .Represent T by an operator matrix with respect to this decomposition: One says that an operator X ∈ B(K) is a dilation of an operator T ∈ B(H) if H is a closed subspace of K and, for every positive integer n, T n is the compression to H of X n , or, in other words, if T n = P H X n |H for all n ≥ 1.The preceding paragraph shows that, if T has the block upper triangular form (13.2) with respect to the orthogonal decomposition According to a lemma of D. Sarason [19, Lemma 0], a converse statement also holds: Lemma 13.3.Let X be an operator on a Hilbert space K and suppose X is a dilation of an operator T on a closed subspace H of K. Then there exist closed subspaces H 1 and H 3 of K such that K is the orthogonal direct sum of H 1 , H and H 3 , X has an upper triangular operator matrix with respect to the decomposition K = H 1 ⊕ H ⊕ H 3 , and T is the (2, 2) block in the operator matrix of X.
In the above statement of the lemma, the triangularity assertion about X is equivalent to both H 1 and H 1 ⊕ H being invariant subspaces of K for X.Here is a proof (which Sarason attributes to C. Foias) of Lemma 13.3.
Proof.Let M be the smallest closed X-invariant subspace of K containing H and let P and Q be the orthogonal projections from K onto H and M respectively.Let H 1 = M⊖H, so that Q − P is the orthogonal projection onto H 1 .We assert that H 1 is X-invariant, or equivalently that (Q − P )X(Q − P ) = X(Q − P ).Since clearly QXQ = XQ and QXP = XP , it is enough to show that P XP = P XQ.Now for y ∈ H and for m, n ≥ 1, the dilation condition implies that P X n P X m y = P X n T m y = T n T m y = T n+m y = P X n+m y.
But M is the closed span of the vectors X m y with y ∈ H and m ≥ 1.We may conclude that P X n P x = P X n x for all x ∈ M, and so P XP = P XQ, as required.Thus H 1 and M are invariant subspaces for X.Choice of H 3 = M ⊥ completes the orthogonal decomposition K = H 1 ⊕H⊕H 3 with respect to which X has an upper triangular operator matrix, with T in the (2, 2)-position.
In the theory of operators on Hilbert space the most important result concerning dilations is the Nagy dilation theorem, which asserts that any contraction T on a Hilbert space H has a dilation U on some Hilbert space K ⊇ H such that U is a unitary operator on K.There are many proofs of this result (see [14, page 52]).One of the shortest proofs is to deduce it from the following generalization of the Herglotz Representation Theorem due to Naimark [17,16].and on equating coefficients of z n we deduce that T n = I * U n I for all positive integers n, so that U is a dilation of T .The lemma of Sarason proved above then gives us a geometric interpretation of the relation between U and T : T is the compression of U to a semi-invariant subspace of U, which is to say, T is the compression of U to the orthogonal complement of of one invariant subspace of U in another.In the notation of the above proof of Sarason's Lemma, T is the compression of U to M ⊖ H 1 , while both M and H 1 are invariant subspaces for U, which makes H = M ⊖ H 1 a semi-invariant subspace of U.

3 .
condition for W (T ) to be in an ellipse 8 Strange dilation on an ellipse 10 1. Introduction

3 .By [ 12 ,
Theorem 1.2-1], the spectrum σ(T ) of an operator T is contained in W (T ), and so, by the Riesz-Dunford functional calculus, ϕ(T ) is defined for all ϕ ∈ Hol(C) and T ∈ F bfd (C).Recall further from [2,Chapter 9], that the Douglas-Paulsen family F dp (δ) corresponding to the annulus R δ is the class of Douglas-Paulsen operators X with parameter δ that satisfy the additional condition σ(X) ⊆ R δ .For Ω an open set in the plane, Hol(Ω) will denote the set of holomorphic functions defined on Ω and H ∞ (Ω) will denote the set of bounded holomorphic functions defined on Ω, with the supremum norm ϕ ∞ = sup z∈Ω |ϕ(z)|.Following the setup of [2, Chapter 9], we consider the calcular norms ϕ dp = sup and Y ∈ B(L) by Y = P L U|L, then p(Y ) = P L p(U)|L (5.5)

11 .
The bfd norm on Hol(G δ ) and the dp norm on Hol(R δ ) Following[2, Chapter 9], for any δ ∈ (0, 1), we define the Douglas-Paulsen family with parameter δ to be the class F dp (δ) of all Douglas-Paulsen operators T with parameter δ such that σ(T ) ⊆ R δ and consider the associated calcular normϕ dp = sup T ∈F dp (δ) ϕ(T ) for ϕ ∈ Hol(R δ ).(11.1) Also following [2, Chapter 9] we define the B. and F. Delyon family F bfd (G δ ) corresponding to the elliptical region G δ to be the class of operators T such that W (T ) ⊆ G δ .By [12, Theorem 1.2-1], the spectrum σ(T ) of an operator T is contained in W (T ), and so, by the Riesz-Dunford functional calculus, ϕ(T ) is defined for all ϕ ∈ Hol(G δ ) and T ∈ F bfd (G δ ).Consider the calcular norm

Theorem 13 . 4 .
Let H be a Hilbert space and assume that V is an analytic B(H)-valued function defined on D satisfying Re V (z) ≥ 0 for all z ∈ D. Then there exist a Hilbert space K, an isometry I : H → K, and a unitary operator U ∈ B(K) such thatV (z) = I * 1 + zU 1 − zU I for all z ∈ D.Indeed, if T is a contraction then, for any z ∈ D,Re 1 + zT 1 − zT = 2(1 − zT ) −1 (1 − |z| 2 T T * )(1 − zT * ) −1 > 0.It follows from Naimark's theorem that there exist a Hilbert space K, an isometry I : H → K, and a unitary operatorU ∈ B(K) such that 1 + zT 1 − zT = I * 1 + zU 1 − zU I for all z ∈ D.Expanding both sides of the equation as power series in z, we have1 + 2zT + 2z 2 T 2 + 2z 3 T 3 + • • • = I * (1 + 2zU + 2z 2 U 2 + 2z 3 U 3 + . . .)I for all z ∈ D, Re G(z) ≥ 0. Take the composition of the inverse C −1 of the Cayley transform C : D → H with G, where H denotes the right half plane {z ∈ C : Re z > 0}.Then C −1 • G is an analytic B(H)-valued function on D, continuous on D and maps T to the unit ball of B(H).By the Maximum Principle, for all z ∈ D, .2) Theorem 8.3.Let T ∈ B(H) be generic for K δ .Then T ∈ W δ (H) if and only if there exists an invertible S ∈ B(H) such that 12is the 1-2 entry of C when C is represented as a 2 × 2 block matrix acting on H ⊕ H.To see the relation (8.10) note that equation(8.6)implies that C 12 = QS −1 and that if Q is not invertible, then 4δ ∈ σ(T 2 ), contradicting the genericity of T .In light of the above properties of C we make the following definition.Definition 8.11.Let δ ∈ (0, 1).If H is a finite-dimensional Hilbert space and dim H = n, we say that C is a (δ, H)-germinator if C ∈ B(H ⊕ H) and C satisfies the conditions: 12 is the 1-2 entry of C when C is represented as a 2 × 2 block matrix acting on H ⊕ H.
Proof.By Proposition 8.16, if T ∈ W δ (H) is generic for K δ , then exists an invertible S ∈ B(H) such that if C is defined by the equation .21)Evidently A k and Y k contractive for all k imply that A and Y are contractions.Similarly, as A k is self-adjoint for all k, A is self-adjoint.Finally, equation (9.20) and relations (9.21) imply equation(9.19).Note that if we take δ = 0 in equation 9.19 then the statement of Theorem 9.18 formally becomes the following.Let T ∈ B(H) where H is a finite dimensional Hilbert space.Then W (T ) ⊆ D if and only if there exist a pair of contractions A, Y ∈ B(H) such that A is self-adjoint and Lemma 11.13.Let C be a smoothly bounded open strictly convex set in C and ε > 0.Proposition 11.16.Let δ ∈ (0, 1) and let ϕ ∈ Hol(G δ ).The following equation holds: bbc (c, r).(11.15)It is easy to see that G δ is indeed a smoothly bounded open strictly convex set in C.
T 11 T 12 T 13 T 21 T 22 T 23 T 31 T 32 T 33Here of course T ij denotes the operator P i T |H j , where P i is the orthogonal projection from H to H i and •|H j denotes restriction to H j .In general the properties of the block entries T ij are related in quite subtle ways to the properties of the operator T , but in the special case that the block matrix (13.1) is lower triangular, in the sense that the operators T 21 , T 31 , T 32 are all zero (or equivalently, that H 1 and H 1 ⊕ H 2 are invariant subspaces of H for T ), the relation is more straightforward.Observe in particular that, if T has a block upper triangular matrix with respect to the orthogonal decompositionH = H 1 ⊕ H 2 ⊕ H 3 , saywhere the stars denote unspecified entries.Thus, in this block triangular case, (T 22 ) n is the compression of T n to H 2 for all n ≥ 1: T n 22 = P 2 T n |H 2 .This property brings us to the terminology of dilations.