Traces of certain integral operators related to the Riemann hypothesis

: We prove the existence of a nontrivial singular trace τ deﬁned on an ideal J closed with respect to the logarithmic submajorization such that τ ( A ρ ( α )) = 0, where A ρ ( α ) ∶ L 2 ( 0 , 1 ) → L 2 ( 0 , 1 ) , [ A ρ ( α ) f ]( θ ) = ∫ 1 0 ρ ( αθ / x ) f ( x ) dx , 0 < α ≤ 1. We also show that τ ( A ρ ( α )) = 0 for every τ nontrivial singular trace on J . Finally, we give a recursion formula from which we can evaluate all the traces Tr ( A r ρ ( α )) , r ∈ N , r ≥ 2.


Introduction
Let B(H) be the algebra of all bounded linear operators on a separable complex Hilbert space H. The adjoint of an operator T ∈ B(H) is denoted by T * and the symbol I stands for identity maps. Denote by {s n (T )} n≥1 the sequence of singular values of a compact operator T ∈ B(H). If 0 < p < ∞ we say that T ∈ S p (H) if ∑ ∞ n=1 s p n (T ) < ∞; the set S p (H) is a two-side ideal in B(H). The sets S 1 (H) and S 2 (H) will denote, respectively, the set of nuclear and Hilbert-Schmidt operators. By an ideal we mean a two-sided ideal in B(H). A linear functional τ from the ideal J into C is said to be a trace if: i) τ(U * T U) = τ(T ) for every T ∈ J and U ∈ B(H) unitary. Equivalently, τ(S T ) = τ(T S ) for every T ∈ J and S ∈ B(H). ii) τ(T ) ≥ 0 for every T ∈ J with T a non-negative operator. We denote by T ≥ 0 when T is a non-negative operator.
Then a trace is a positive unitarily invariant linear functional. Let T ∈ S 1 (H), and let {ϕ n } n≥1 be an orthonormal system in H. By [13, p. 56], we have n j=1 ⟨T ϕ j , ϕ j ⟩ ≤ n j=1 s j (T ) , ∀n ≥ 1. Therefore, ⟨T ϕ n , ϕ n ⟩ is well-defined, since the right-hand series converges absolutely, and its value does not depend on the choice of the orthonormal basis {ϕ n } n≥1 . Clearly, this linear functional is a trace on the ideal S 1 (H).
There is also the description of Tr as the sum of eigenvalues, where {λ n (T )} n≥1 is the sequence of nonzero eigenvalues of T , ordered in such a way that λ n (T ) ≥ λ n+1 (T ) , ∀ n ∈ N and each one of them being counted according to its algebraic multiplicity. This result was shown by Von Neumann in [26] for self-adjoint operators and by Lidskii in [19] in general case. Formula (1.1) is called the Lidskii formula. A natural question concerning the extension of the Lidskii formula to other ideals and traces on these ideals has been addressed in [6,13,[20][21][22]. Thus, the notion of spectral traces arises. A trace τ on an ideal J is called spectral if for every T ∈ J, the value of τ(T ) depends only on the eigenvalues of T and their multiplicities. For example, the classical trace Tr on the ideal S 1 (H) is spectral. Motivated by the problem of identifying spectral traces, it is shown in [12,Corollary 2.4] that every trace on a geometrically stable ideal is spectral. This result has been generalized in the setting of ideals closed with respect to the logarithmic submajorization (see [24]).
A trace τ on an ideal J will be called singular if it vanishes on the set F(H) of finite rank operators. This definition makes sense, since by the Calkin Theorem [9], each proper ideal in B(H) contains the finite rank operators and is contained in the ideal K(H) of the compact linear operators on H.
In 1966, J. Dixmier proved the existence of singular traces [11]. These traces are called Dixmier traces, and its importance is due to their applications in noncommutative geometry [10]. Other examples of singular traces appeared in [1,17,25].
The question whether an operator belongs to the domain of some singular trace was answered in [15]. Motivated by this, we give a nontrivial singular trace taking the value of zero on certain integral operator related to the Riemann hypothesis.
In [2], J. Alcántara-Bode has reformulated the Riemann hypothesis as a problem of functional analysis by means of the following theorem.
Observe that A ρ (1) = A ρ . Since A ρ is nonnuclear (see [2,Theorem 6]), it was proved in [23,Theorem 4.3] that there exists a nontrivial singular trace τ with domain a geometrically stable ideal J such that A ρ ∈ J and τ(A ρ ) = 0. It follows from [24,Lemma 35] that J is closed with respect to the logarithmic submajorization.
Let J be as in the previous paragraph. We show that if 0 < α ≤ 1 then A ρ (α) ∈ J , and that each nontrivial singular trace on J takes the value of zero on A ρ (α). More precisely, our first main result of the paper is the following theorem. Theorem 1.2. If 0 < α ≤ 1 then τ(A ρ (α)) = 0 for every τ nontrivial singular trace on J , where J is the geometrically stable ideal in the above paragraph.
The approach used to prove Theorem 1.2 is based on spectral traces and the fact that the operators α(A ρ + Q f 1 ) and A ρ + Q fα , where α ∈]0, 1[, have the same nonzero eigenvalues with the same algebraic and geometric multiplicities.
Finally, our second main result is a recursion formula to calculate the traces Tr (A r ρ (α)), r ∈ N, r ≥ 2.
Theorem 1.3. If 0 < α ≤ 1 then for every r ∈ N with r ≥ 2 we have In order to prove Theorem 1.3, we use the modified Fredholm determinant of I − uA ρ (α) (see [3]).

Singular traces
Let l ∞ the space of all bounded sequence of complex numbers and w a dilation invariant extended limit on l ∞ , that is, w is an extended limit on l ∞ and It was shown in [11] that the weight Tr w defines a positive, unitarily invariant, additive and positive homogeneous function on the positive cone of M 1,∞ (H), that can uniquely be extended to a singular trace on all of M 1∞ (H), i.e., for an arbitrary T ∈ M 1,∞ (H), its Dixmier trace is defined by In addition to this, the Dixmier trace vanishes on the ideal S 1 (H) and is continuous in the norm ⋅ 1,∞ .
The existence of a singular trace which is nontrivial on a compact operator T , i.e., on the two-sided ideal generated by T , was studied by J. Varga [25], and it has been completely characterized in [1]. This leads to study irregular, eccentric and generalized eccentric operators.
In this context, the main result in [1] is the following. The process to construct the singular trace given by a) is as follows: We introduce a triple Ω = (T, w, {n k } k≥1 ), where T is a generalized eccentric operator, w is an extended limit and n k = kp k , k ∈ N, where {p k } k≥1 is the sequence given in Lemma 2.3. Associated with the triple Ω, on the positive part of the ideal (T ), we defined the functional and by [1,Theorem 2.11], this functional extends linearly to a singular trace on the ideal (T ).
a) By Theorem 2.4, finite rank operators cannot by generalized eccentric. b) The question whether an operator belongs to the domain of some singular trace is treated in [15]. By [15, Theorem 3.1 (i)], every compact operator A is in the kernel (hence in the domain) of some singular trace. The main idea for proving this theorem is the existence of a generalized eccentric operator B such that A ∈ (B) 0 ⊂ (B). Here (B) 0 denotes the kernel of (B) (see [15]). By [14, p. 172], s n (Re(V)) = 2 (2n − 1)π , n = 1, 2, . . ., it follows that Re(V) is a generalized eccentric operator. However, the operator Im(V) has rank one, and by Remark 2.5 (a), Im(V) is not an eccentric operator.

Commutator subspace
Now we concentrate on the commutator subspace of geometrically stable ideals and ideals closed with respect to the logarithmic submajorization, terminologies used in [18,24], respectively.  The following theorem [18,Theorem 3.3] characterizes in terms of arithmetic means the commutator subspace of geometrically stable ideals.

Modified Fredholm determinant
If T ∈ S 2 (H) and {λ n (T )} n≥1 is the sequence of non-zero eigenvalues of T , each repeated according to its algebraic multiplicity and ordered in such a way that λ n (T ) ≥ λ n+1 (A) , ∀n ∈ N then det 2 (I − λT ) =   of I − λT is equal to its Hilbert-Carleman determinant (see [13, p. 176]). More precisely,

The integral operator A ρ
To study the Riemann hypothesis, J. Alcántara-Bode [2] introduced the integral operator As we have seen in the introduction, the operators A ρ (α) arise from the attempt to verify the condition h ∈ Ran (A ρ ).

Main results
For a given compact operator T ∉ S 1 (H), where H is a separable complex Hilbert space, the following theorem shows the existence of a nontrivial singular trace defined on an ideal closed with respect to the logarithmic submajorization taking the value of zero on T . that extends linearly to a singular trace on the ideal (B). We also denote this extension by t Ω . Clearly t Ω ( B ) = 1. Since T ∉ S 1 (H), it follows that B ∉ S 1 (H) and then It follows from (6.1) that t Ω is bounded with respect to the norm ⋅ B , where Since A ρ ∉ S 1 (L 2 (0, 1)), it follows from Theorem 6.1 that there exists an ideal J of B(L 2 (0, 1)) closed with respect to the logarithmic submajorization that contains A ρ and a nontrivial singular trace τ on J such that τ(A ρ ) = 0. The following theorem shows that every nontrivial singular trace on J take the value of zero on A ρ . Theorem 6.2. τ(A ρ ) = 0 for every τ nontrivial singular trace on J .
Proof. Let τ be a nontrivial singular trace on J (since A ρ ∉ S 1 (L 2 (0, 1)), the existence of J and a nontrivial singular trace on J are guaranteed by Theorem 6.1. If 0 < α < 1, by [5], the operators α(A ρ + Q f 1 ) and A ρ + Q fα have the same nonzero eigenvalues with the same algebraic and geometric multiplicities, where Q f (g) = ⟨g, f ⟩h, . It follows from Theorem 3.8 that τ is a spectral trace and, therefore, Hence, by definition of singular trace, it follows from (6.2) that τ(A ρ ) = 0.

Remark 6.3.
i) It follows from Theorem 3.8 that the singular trace given in Theorem 6.1 is spectral. ii) If 0 < α < 1, by [5], the kernel of the operators α(A ρ + Q f 1 ) and A ρ + Q fα is trivial if and only if the Riemann hypothesis is true.
The operator V α was introduced in [4]. We have from (6.4) that A ρ (α) ∈ J for every α ∈ J .
We are now ready to prove the first main result of the paper.
Finally, using the modified Fredholm determinant of I − uA ρ (α), a recursion formula is presented to calculate all the traces Tr (A r ρ (α)), r ≥ 2.

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