Some remarks on spectra of nuclear operators

It was shown by M. I. Zelikin (2007) that the spectrum of a nuclear operator in a separable Hilbert space is central-symmetric iff the spectral traces of all odd powers of the operator equal zero. The criterium can not be extended to the case of general Banach spaces: It follows from Grothendieck-Enflo results that there exists a nuclear operator $U$ in the space $l_1$ with the property that $\operatorname{trace}\, U=1$ and $U^2=0.$ B. Mityagin (2016) has generalized Zelikin's criterium to the case of compact operators (in Banach spaces) some of which powers are nuclear. We give sharp generalizations of Zelikin's theorem (to the cases of subspaces of quotients of $L_p$-spaces) and of Mityagin's result (for the case where the operators are not necessarily compact).


Introduction
It was shown by M. I. Zelikin in [1] that the spectrum of a nuclear operator in a separable Hilbert space is central-symmetric if and only if the spectral traces of all odd powers of the operator equal zero. Recall that the spectrum of every nuclear operator in a Hilbert space consists of non-zero eigenvalues of nite algebraic multiplicity, which have no limit point except possibly zero, and maybe zero. This system of all eigenvalues (written according to their multiplicities) is absolutely summable, and the spectral trace of any nuclear operator is, by de nition, the sum of all its eigenvalues (taken according to their multiplicities).
The space of nuclear operators in a Hilbert space may be de ned as the space of all trace-class operators (see [2, p. 77]); in this case we speak about the "nuclear trace" of an operator). Trace-class operators in a Hilbert space can be considered also as the elements of the completion of the tensor product of the Hilbert space and its Banach dual with respect to the greatest crossnorm on this tensor product [2, p. 119]. The wellknown Lidskiǐ theorem [3] says that the nuclear trace of any nuclear operator in a Hilbert space (or, what is the same, of the corresponding tensor element) coincides with its spectral trace. Thus, Zelikin's theorem [1] can be reformulated in the following way: the spectrum of a nuclear operator in a separable Hilbert space is central-symmetric if and only if the nuclear traces of all odd powers of the corresponding tensor element are zero.
One of the aim of our notes is to give an exact generalization of this result to the case of tensor elements of so-called s-projective tensor products of subspaces of quotients of L p (µ)-spaces. In particular, we get as a consequence Zelikin's theorem (taking p = ).
Another problem which is under consideration in our notes is concentrated around the so-called Z dsymmetry of the spectra of the linear operators. The notion of Z d -symmetry of the spectra was introduced by B. S. Mityagin in a preprint [4] and in his paper [5]. He is interested there in a generalization of the result from [1] in two directions: to extend Zelikin's theorem to the case of general Banach spaces and to change the property of a compact operator to have central-symmetric spectrum to have Z d -symmetric spectrum. Roughly speaking, Z d -symmetry of a spectrum of a compact operator T means that for any non-zero eigenvalue λ of T the spectrum contains also as eigenvalues of the same algebraic multiplicities all "d-shifted" numbers tλ . B. S. Mityagin has obtained a very nice result, showing that the spectrum of a compact operator T in an arbitrary Banach space, some power T m of which is nuclear, is Z d -symmetric if and only if for all large *Corresponding Author: Oleg I. Reinov: Saint Petersburg State University, E-mail: orein51@mail.ru enough integers of type kd + r ( < r < d) the nuclear traces of T kd+r are zero. We present some thoughts around this theorem, giving, in particular, a short (but not so elementary as in [4,5]) proof for the case where the operator is not necessarily compact. Let us mention, however, that the proof from [4,5] can be adapted for this situation too.
Some words about the content of the paper. In Section 2, we introduce some notation, de nitions and terminology in connection with so-called sprojective tensor products, s-nuclear operators and the approximation properties of order s, s ∈ ( , ]. We formulate here two auxiliary assertions from the paper [6]; they give us generalized Grothendieck-Lidskiǐ trace formulas which will be useful in the next section.
Section 3 contains an exact generalization of Zelikin's theorem. In this section we present a criterion for the spectra of s-nuclear operators in subspaces of quotients of L p -spaces to be central-symmetric.
Results of Section 4 show that the criterion of the central symmetry, obtained in the previous section, is optimal. In particular, we present here (Theorem 4.1) sharp examples of s-nuclear operators T in the spaces l p , ≤ p ≤ +∞, p ≠ , for which trace T = and T = .
Finally, Section 5 is devoted to the study of Mityagin's Z d -symmetry of the spectra of linear operators. Our aim here is to give a short (but using the Fredholm Theory) proof of Mityagin's theorem [4,5] for arbitrary linear continuous (Riesz) operators. Firstly, we consider a Z situation (central symmetry) to clarify an idea which is to be used then in the general case. We nish the paper with a short proof of the theorem from [4,5] for continuous (not necessarily compact) operators and with some simple examples of applications.

Preliminaries
By X, Y , . . . we denote the Banach spaces, L(X, Y) is a Banach space of all linear continuous operators from X to Y; L(X) ∶= L(X, X). For a Banach dual to a space X we use the notation X * . If x ∈ X and x ′ ∈ X * , then ⟨x ′ , x⟩ denotes the value x ′ (x). By X * ⊗ X we denote the projective tensor product of the spaces X * and X [7] (see also [8,9]). It is a completion of the algebraic tensor product X * ⊗ X (considered as a linear space of all nite rank continuous operators w in X) with respect to the norm Every element u of the projective tensor product X * ⊗ X can be represented in the form . More generally, if < s ≤ , then X * ⊗ s X is a subspace of the projective tensor product, consisting of the tensor elements u, u ∈ X * ⊗ X, which admit representations of the form On the linear space X * ⊗ X, a linear functional "trace" is de ned in a natural way. It is continuous on the normed space (X * ⊗ X, ⋅ ∧ ) and has the unique continuous extension to the space X * ⊗ X, which we denote by trace . Every This de nes a natural mapping j ∶ X * ⊗ X → L(X). The operators, lying in the image of this map are called nuclear [7,10] then the corresponding operatorũ is called s-nuclear [11,12]. By j s we denote a natural map from X * ⊗ s X to L(X). We say that a space X has the approximation property of order s, < s ≤ (the AP s ), if the canonical mapping j s is one-to-one [11,12]. Note that the AP is exactly the approximation property AP of A. Grothendieck [2,8]. Classical spaces, such as L p (µ) and C(K), have the approximation property. If a space X has the AP s , then we can identify the tensor product X * ⊗ s X with the space N s (X) of all s-nuclear operators in X (i.e. with the image of this tensor product under the map j s ). In this case for every operator T ∈ N s (X) = X * ⊗ s X the functional trace T is well de ned and called the nuclear trace of the operator T.
It is clear that if a Banach space has the approximation property, then it has all the properties AP s , s ∈ ( , ]. Every Banach space has the property AP (A. Grothendiek [7], see also [11]). Since each Banach space is a subspace of an L ∞ (µ)-space, the following fact (to be used below) is a generalization of the mentioned result of A. Grothendieck: If a Banach space Y is isomorphic to a subspace of a quotient (or to a quotient of a subspace) of some L p (µ)-space, then it has the AP s .
Thus, for such spaces we have an equality Y * ⊗ s Y = N s (Y) and the nuclear trace of any operator T ∈ N s (Y) is well de ned.
We will need also the following auxiliary assertion (the rst part of which is a consequence of the previous lemma).

the system of all eigenvalues of the operator T (written according to their algebraic multiplicities), and
Following [1], we say that a spectrum of a compact operator in a Banach space is central-symmetric, if for each of its eigenvalue λ the number −λ is also its eigenvalue and of the same algebraic multiplicity. We shall use the same terminology in the case of operators, all non-zero spectral values which are eigenvalues of nite multiplicity and have no limit point except possibly zero; the corresponding eigenvalue sequence for such an operator T will be denoted by sp (T); thus it is an unordered sequence of all eigenvalues of T taken according to their multiplicities.

On central symmetry
Let us note rstly that the theorem of Zelikin (in the form as it was formulated in [1]) can not be extended to the case of general Banach spaces, even if the spaces have the Grothendieck approximation property. Example 3.1. Let U be a nuclear operator in the space l , constructed in [10,Proposition 10.4.8]. This operator has the property that trace U = and U = . Evidently, the spectrum of this operator is { }. Let us note that the operator is not only nuclear, but also belongs to the space N s (l ) for all s ∈ ( , ]. It is not possible to present such an example in the case of -nuclear operators (see Corollary 3.6 below). Note also that, however, the traces of all operators U m , m = , , . . . , (in particular, U n− ) are equal to zero.  Since under the conditions of Theorem 3.3 the space Y has the AP s , the tensor product Y * ⊗ s Y can be identi ed naturally with the space of all s-nuclear operators in Y . Hence, the statement of Theorem 3.3 may be reformulated in the following way:

be a subspace of a quotient (or a quotient of a subspace) of an L p -space, T be an s-nuclear operator in Y . The spectrum of T is central-symmetric if and only if trace T n−
= , n ∈ N.

Corollary 3.5 ([1]). The spectrum of a nuclear operator T, acting on a Hilbert space, is central-symmetric if and only if trace T n−
For a proof, it is enough to apply Theorem 3.3 for the case p = .

Corollary 3.6. The spectrum of a 2/3-nuclear operator T, acting on an arbitrary Banach space, is centralsymmetric if and only if trace T n−
For a proof, it is enough to apply Theorem 3.3 for the case p = ∞, taking into account the fact that every Banach space is isometric to a subspace of an L ∞ (µ)-space.
In connection with Corollary 3.6, let us pay attention again to the nuclear operator from Example 3.1.

Sharpness of results of Section 3
Now we will show that the statement of Theorem 3.3 is sharp and that the exponent s is optimal if p is xed (if of course p ≠ , i.e. s ≠ ). Consider the case < p ≤ ∞. In this case s = + − p = − p. In a paper of the author [9, Example 2] the following result was obtained (see a proof in [9]): There exist a subspace Y p of the space l p (c if p = ∞) and a tensor element w p ∈ Y * p⊗ Y p such that w p ∈ Y * p⊗s Y p for every s > r, trace w p = ,w p = and the space Y p (as well as Y * p ) has the AP r (but evidently does not have the AP s if ≥ s > r). Moreover, this element admits a nuclear representation of the form Evidently, we have for a tensor element u ∶= w p from the assertion (⋆) ∶ trace u = and the spectrum of the operatorũ equals { }. The case where < p ≤ ∞ can be considered analogously (with an application of the assertion (⋆) to a "transposed" tensor element w t p ∈ Y p⊗ Y * p .) As was noted above (Example 3.1), there exists a nuclear operator U in l such that U = and trace U = .
The following theorem is an essential generalization of this result and gives us the sharpness of the statement of Corollary 3.4 (even in the case where Y = l p ).
Proof. Suppose that p > . Consider the tensor element w ∶= w p from the assertion (⋆) and its representation Let l ∶ Y ∶= Y p → l p be the identity inclusion. Let y ′ k be an extension of the functional x ′ k (k = , , . . . ) from the subspace Y to the whole space l p with the same norm and set v ∶= ∑ ∞ k= µ k y ′ k ⊗ l(x k ). Then v ∈ l p ′⊗ s l p ( p + p ′ = ) for each s ∈ (r, ], trace v = ∑ µ k ⟨y ′ k , l(x k )⟩ = andṽ(l p ) ⊂ l(Y) ⊂ l p . On the other hand, we have a diagram: whereṽ is an operator generated byṽ,ṽ = lṽ andṽ l =w = . Put V ∶=ṽ. Clearly, trace V = and the spectrum sp V = { }. Let us note that V ∉ N r (l p ) (by Lemma 2.2). If p ∈ [ , ), then it is enough to consider the adjoint operator.
It follows from Theorem 4.1 that the assertion of Corollary 3.4 is optimal already in the case of the space Y = l p (which, by the way, has the Grothendieck approximation property).

Generalizations: around Mityagin's theorem
Recall that if T ∈ L(X) and, for some m ∈ N, T m is a Riesz operator (see, e.g., [15, p. 943] for a de nition), then T is a Riesz operator too (see, e.g., [12, 3.2 We are going to present a short proof of the theorem of B. Mityagin from [4,5]. To clarify our idea of the proof, let us consider rstly the simplest case where d = .

Theorem 5.1. Let X be a Banach space and T ∈ L(X). Suppose that some power of T is nuclear. The spectrum of T is central-symmetric if and only if there is an integer K ≥ such that for every l > K the value trace T l is well de ned and trace T l+
= for all l > K.
Proof. Suppose that T ∈ L(X) and there is an M ∈ N so that T M ∈ N(X). Fix an odd N , N > M, with the property that T N ∈ N (X) (it is possible since a product of three nuclear operators is 2/3-nuclear) and trace T N + k = for all k = , , , . . . . By Corollary 3.6, the spectra of all T N + k are central-symmetric (since, e.g., trace T N = trace (T N ) = trace (T N ) = ⋅ ⋅ ⋅ = by assumption). Assume that the spectrum of T is not central-symmetric. Then there exists an eigenvalue λ ∈ sp (T) such that −λ ∉ sp (T).
Hence, there exist µ N ∈ sp (T) and θ N so that θ N = , Hence, there exist µ N + ∈ sp (T) and θ N + so that θ N + = , µ N + N + = −λ N + and µ N + = θ N + λ , θ N + ≠ − etc. By induction we get the sequences {µ N + k } ∞ k= and {θ N + k } ∞ k= with the properties that µ N + k ∈ sp (T), Now we are going to consider a general case of a notion of Z d -symmetry of a spectra, introduced and investigated by B. Mityagin in [4,5]. Let T be an operator in X, all non-zero spectral values which are eigenvalues of nite multiplicity and have no limit point except possibly zero. Recall that we denote by sp (T) the corresponding unordered eigenvalue sequence for T (possibly, including zero). For a xed d = , , . . . and for the operator T, the spectrum of T is called Lemma 5.2. Let Φ(X) be a linear subspace of X * ⊗ X of spectral type l , i.e., for every v ∈ Φ(X) the series Proof. If the function det ( − zu) is d-even, then the eigenvalue sequence ofũ is Z d -symmetric, since this sequence coincides with the sequence of inverses of zeros of det ( − zu) (according to their multiplicities).
If the eigenvalue sequence ofũ is Z d -symmetric, then trace u = ∑ λ∈sp (ũ) λ = (since Φ(X) is of spectral type l and ∑ t∈ d √ t = ). Also, by the same reason trace u kd+r = for all k = , , , . . . and r = , , . . . , d− , since the spectrum ofũ l is absolutely summable for every l ≥ and we may assume that ) in a neighborhood U of zero. Therefore, this function is d-even in the neighborhood U. By the uniqueness theorem det ( − zu) is d-even in C. Now we are ready to present a short proof of the theorem of B. Mityagin [4,5]. Note that the theorem in [4,5] is formulated and proved for compact operators, but the proof from [4,5] can be easily adapted for the general case of linear operators.
Theorem 5.5. Let X be a Banach space and T ∈ L(X). Suppose that some power of T is nuclear. The spectrum of T is Z d -symmetric if and only if there is an integer K ≥ such that for every l > Kd the value trace T l is well de ned and trace T kd+r = for all k = K, K + , K + , . . . and r = , , . . . , d − .
Proof. Fix N ∈ N such that T N is -nuclear (it is possible by a composition theorem from [7, Chap, II, Theor. 3, p. 10]). Note that, by A. Grothendieck, the trace of T l is well de ned for all l ≥ N .
Suppose that the spectrum of T is Z d -symmetric. Take an integer l ∶= kd + r ≥ N with < r < d. Since the spectrum of T l is absolutely summable, trace T l = ∑ λ∈sp (T l ) λ and we may assume that {λ m (T l )} = {λ m (T) l }, we get that trace T kd+r = . In proving the converse, we may (and do) assume that Kd > N . Consider an in nite increasing sequence {p m } of prime numbers with p > (K + )d. Assuming that trace T kd+r = if k = K, K + , K + , . . . and r = , , . . . , d − , for a xed p m we get from Lemma 5.2 (more precisely, from Corollary 5.3) that the function det ( − zT p m ) is d-even. Suppose that the spectrum of T is not Z d -symmetric. Then there exist an eigenvalue λ ∈ sp (T) and a root θ ∈ d √ so that θλ ∉ sp (T). On the other hand, again by Lemma 5.2, the spectrum of T p m is Z d -symmetric. Since λ p m ∈ sp (T p m ), there exists µ m ∈ sp (T) such that µ p m m = θ p m λ p m ∈ sp (T p m ); hence, µ m = θ m λ for some θ m with θ m = . But λ > . Therefore, the set {µ m } is nite and it follows that there is an integer M > such that θ M = θ M+ = θ M+ = . . . . Hence, θ p m = θ p m M for all m ≥ M. Thus, θ M = θ. A contradiction.
Let us give some examples in which we can apply Theorem 5.5, but the main result of [4,5] does not work.
Example 5.6. Let Π p be the ideal of absolutely p-summing operators (p ∈ [ , ∞); see [10] for a de nition and related facts). Then for some n one has Π n p ⊂ N. In particular, Π (C[ , ]) ⊂ N(C[ , ]), but not every absolutely 2-summing operator in C[ , ] is compact. Another interesting example: Π is of spectral type l [16]. We do not know (maybe it is unknown to everybody), whether the nite rank operators are dense in this ideal. However, Theorem 5.5 may be applied. Moreover, it can be seen that, for example, the spectrum of an operator T from Π is central-symmetric if and only if the spectral traces of the operators T k− are zero for all k > .