Some remarks on spectra of nuclear operators

Oleg I. Reinov

doi:10.1515/math-2018-0043

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Some remarks on spectra of nuclear operators

Oleg I. Reinov

Published/Copyright: April 26, 2018

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$Open Mathematics$

From the journal Open Mathematics Volume 16 Issue 1

Abstract

We give criteria for the spectra of some nuclear operators in subspaces of quotients of L_p-spaces to be central-symmetric, as well as for the spectra of linear operators in Banach spaces to be Z_d-symmetric in the sense of B. Mityagin. Also, we present a short proof of a corresponding Mityagin’s theorem.

Keywords: Tensor product; Approximation property; Eigenvalue; Fredholm determinant

MSC 2010: 47B06; 47B10

1 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 finite 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 definition, the sum of all its eigenvalues (taken according to their multiplicities).

The space of nuclear operators in a Hilbert space may be defined 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 well-known 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 = 2).

Another problem which is under consideration in our notes is concentrated around the so-called ℤ_d-symmetry 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 for t∈1d. 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 enough integers of type kd + r (0 < 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, definitions and terminology in connection with so-called s-projective tensor products, s-nuclear operators and the approximation properties of order s, s ∈ (0, 1]. 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, 1 ≤ p ≤ + ∞, p ≠ 2, for which trace T = 1 and T² = 0.

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 ℤ₂ situation (central symmetry) to clarify an idea which is to be used then in the general case. We finish 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.

2 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 finite rank continuous operators w in X) with respect to the norm

||w||∧:=inf{∑k=1N||xk′||||xk||:w=∑k=1Nxk′⊗xk}.

Every element u of the projective tensor product X^*⊗̂X can be represented in the form

u=∑iλixi′⊗xi,

where (λ_i) ∈ l₁ and ||xi′||≤1, ∥x_i∥ ≤ 1 [7].

More generally, if 0 < s ≤ 1, then X^*⊗̂_sX is a subspace of the projective tensor product, consisting of the tensor elements u, u ∈ X^*⊗̂X, which admit representations of the form u=∑k=1∞xk′⊗xk, where (xk′) ⊂ X^*, (x_k) ⊂ X and ∑k=1∞||xk′||s||xk||s<∞ [7, 8, 9]. Thus, X^*⊗̂X = X^*⊗̂₁X.

On the linear space X^* ⊗ X, a linear functional “trace” is defined 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 tensor element u, u ∈ X^*⊗̂X, of the form u=∑k=1∞xk′⊗xk generates naturally an operator u͠ : X → X, u͠(x) := ∑k=1∞〈xk′,x〉xk for x ∈ X. This defines a natural mapping j₁ : X^*⊗̂X → L(x). The operators, lying in the image of this map are called nuclear [7, 10]. More generally, if 0 < s ≤ 1, u = ∑k=1∞xk′⊗xk and ∑k=1∞||xk′||s||xk||s<∞, then the corresponding operator u͠ is called s-nuclear [11, 12]. By j_s we denote a natural map from X^*⊗̂_sX to L(x). We say that a space X has the approximation property of order s, 0 < s ≤ 1 (the AP_s), if the canonical mapping j_s is one-to-one [11, 12]. Note that the AP₁ 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^*⊗̂_sX 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^*⊗̂_sX the functional trace T is well defined 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 ∈ (0, 1]. Every Banach space has the property AP_2/3 (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:

Lemma 2.1

([6, Corollary 10]). Let s ∈ (0, 1], p ∈ [1, ∞] and 1/s = 1+|1/p – 1/2|. If a Banach space Y is isomorphic to a subspace of a quotient (or to a quotient of a subspace) of someL_p(μ)-space, then it has theAP_s.

Thus, for such spaces we have an equality Y^*⊗̂_sY = N_s(Y) and the nuclear trace of any operator T ∈ N_s(Y) is well defined.

We will need also the following auxiliary assertion (the first part of which is a consequence of the previous lemma).

Lemma 2.2

([6, Theorem 1]). Let Y be a subspace of a quotient (or a quotient of a subspace) of someL_p(μ)-space, 1 ≤ p ≤ ∞. If T ∈ N_s(Y), where 1/s = 1 + |1/2 – 1/p|, then

the nuclear trace of the operator T is well defined,
∑n=1∞|λn(T)|<∞,where {λ_n(T)} is the system of all eigenvalues of the operator T (written according to their algebraic multiplicities), and
traceT=∑n=1∞λn(T).

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 finite 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.

3 On central symmetry

Let us note firstly 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 l₁, constructed in [10, Proposition 10.4.8]. This operator has the property that trace U = 1 and U² = 0. Evidently, the spectrum of this operator is {0}. Let us note that the operator is not only nuclear, but also belongs to the space N_s(l₁) for all s ∈ (2/3, 1]. It is not possible to present such an example in the case of 2/3-nuclear operators (see Corollary 3.6 below). Note also that, however, the traces of all operators U^m, m = 2, 3, …, (in particular, U^2n–1) are equal to zero.

Remark 3.2

For every nuclear operator T : X → X and for any natural number n > 1, the nuclear trace of Tⁿ is well defined (see [7, Chap. II, Cor. 2, p. 16]) and equals the sum of all its eigenvalues (according to their multiplicity) [7, Chap. II, Cor. 1, p. 15]. Therefore, if the spectrum of a nuclear operator T : X → X is central-symmetric, then for each odd m = 3, 5, 7, … the nuclear trace of the operator T^m is equal to zero. This follows from the fact that the eigenvalue sequences ofT and T^m can be arranged in such a way that {λ_n(T)^m} = {λ_n(T^m)} (see, e.g., [12, 3.2.24, p. 147]).

Let us formulate and prove the central result of this section.

Theorem 3.3

Let Y be a subspace of a quotient (or a quotient of a subspace) of anL_p-space, 1 ≤ p ≤ ∞, and u ∈ Y^*⊗̂_sY, where 1/s = 1 + |1/2 – 1/p|, The spectrum of the operatoru͠ is central-symmetric if and only if trace u^2n–1 = 0, n = 1,2,….

Proof

If the spectrum of u͠ is central-symmetric, then, by Lemma 2.2, trace u = trace u~=∑n=1∞λn(T)=0; also, by Remark 3.2, trace um=∑n=1∞λn(Tm)=0 for m = 3, 5, ….

To prove the converse, we need some information from the Fredholm Theory. Let u be an element of the projective tensor product X^*⊗̂X, where X is an arbitrary Banach space. Recall that the Fredholm determinant det (1 – zu) of u (see [7, Chap. II, p. 13] or [10, 12, 13]) is an entire function

det(1−zu)=1−ztraceu+⋯+(−1)nznαn(u)+…,

all zeros of which are exactly (according to their multiplicities) the inverses of nonzero eigenvalues of the operator u͠, associated with the tensor element u. By [7, Chap. II, Cor. 2, pp. 17-18], this entire function is of the form

det(1−zu)=e−ztraceu∏i=1∞(1−zzi)ezzi,

where {z_i = λ_i(u͠)} is a system of all eigenvalues of the operator u͠ (written according to their algebraic multiplicities). Also, there exists a δ > 0 such that for all z, |z| ≤ δ, we have

det(1−zu)=exp(−∑n=1∞1nzntraceun)(1)

(see [13, p. 350]; cf. [14, Theor. I.3.3, p. 10]).

Now, let u ∈ Y^*⊗̂_sY be as in the formulation of our theorem and suppose that trace u^2n–1 = 0, n = 1, 2, …. By (1), we get: for a neighborhood U = U(0) of zero in ℂ, det (1 – zu) = det (1 + zu) for z ∈ U. Therefore, the entire function det (1 – zu) is even. By definition of det (1 – zu), the sequence of zeros of this function is exactly the sequence of inverses of nonzero eigenvalues of u͠. Hence, the spectrum of u͠ is central-symmetric. □

Since under the conditions of Theorem 3.3 the space Y has the AP_s, the tensor product Y^*⊗̂_sY can be identified 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:

Corollary 3.4

Let s ∈ [2/3, 1], p ∈ [2, ∞], 1/s = 1 + |1/2 – 1/p|, Y be a subspace of a quotient (or a quotient of a subspace) of anL_p-space, T be ans-nuclear operator in Y. The spectrum ofT is central-symmetric if and only if trace T^2n–1 = 0, 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^2n–1 = 0, n ∈ N.

For a proof, it is enough to apply Theorem 3.3 for the case p = 2.

Corollary 3.6

The spectrum of a 2/3-nuclear operator T, acting on an arbitrary Banach space, is central-symmetric if and only if trace T^2n–1 = 0, n ∈ 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.

4 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 fixed (if of course p ≠ 2, i.e. s ≠ 1).

Consider the case 2 < p ≤ ∞. In this case 1/s = 1 + |1/2 – 1/p| = 3/2 – 1/p. In a paper of the author [9, Example 2] the following result was obtained (see a proof in [9]):

(⋆) Let r ∈ [2/3, 1), p ∈ (2, ∞], 1/r = 3/2 – 1/p. There exist a subspace Y_p of the space l_p (c₀if p = ∞) and a tensor elementwp∈Yp∗⊗^1Ypsuch thatwp∈Yp∗⊗^sYpfor every s > r, trace w_p = 1, w̃_p = 0 and the spaceY_p (as well asYp∗) has theAP_r (but evidently does not have the AP_s if 1 ≥ s > r). Moreover, this element admits a nuclear representation of the form

wp=∑k=1∞μkxk′⊗xk,where||xk′||=||xk||=1,∑k=1∞|μk|s<∞∀s>r.

Evidently, we have for a tensor element u := w_p from the assertion (⋆): trace u = 1 and the spectrum of the operator u͠ equals {0}.

The case where 2 < p ≤ ∞ can be considered analogously (with an application of the assertion (⋆) to a “transposed” tensor element wpt∈Yp⊗^1Yp∗.)

As was noted above (Example 3.1), there exists a nuclear operator U in l₁ such that U² = 0 and trace U = 1. 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).

Theorem 4.1

Let p ∈ [1, ∞], p ≠ 2, 1/r = 1 + |1/2 – 1/p|. There exists a nuclear operator V in l_p (in c₀if p = ∞) such that

V ∈ N_s(l_p) for each s ∈ (r, 1];
V ∉ N_r(l_p);
trace V = 1 andV² = 0.

Proof

Suppose that p > 2. Consider the tensor element w := w_p from the assertion (⋆) and its representation w = w=∑k=1∞μkxk′⊗xk, where ||xk′|| = ∥x_k∥ = 1 and ∑k=1∞|μk|s<∞ for each s, s > r. Let l : Y : = Y_p → l_p be the identity inclusion. Let yk′ be an extension of the functional xk′ (k = 1, 2, …) from the subspace Y to the whole space l_p with the same norm and set v:=∑k=1∞μkyk′⊗l(xk). Then v ∈ l_p′⊗̂_sl_p (1/p + 1/p′ = 1) for each s ∈ (r, 1], trace v = ∑μk〈yk′,l(xk)〉=1 and v͠(l_p) ⊂ l(Y) ⊂ l_p. On the other hand, we have a diagram:

Y→llp→v~0Y→llp→v~0Y→v~0lp,

where v͠₀ is an operator generated by v͠, v͠ = lv͠₀ and v͠₀l = w͠ = 0. Put V := v͠. Clearly, trace V = 1 and the spectrum sp V² = {0}. Let us note that V ∉ N_r(l_p) (by Lemma 2.2). If p ∈ [1, 2), 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).

5 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 definition), then T is a Riesz operator too (see, e.g., [12, 3.2.24, p. 147]). In particular, if T^m is compact, then all non-zero spectral values λ(T) ∈ sp (T) are eigenvalues of finite (algebraic) multiplicity and have no limit point except possibly zero. Also, in this case the eigenvalue sequences of T and T^m can be arranged in such a way that {λ_n(T)^m} = {λ_n(T^m)} (see [12, 3.2.24, p. 147]). Recall that in this case we denote by sp (T) (resp., by sp (T^m) the sequence {λ_n(T)^m} (resp., {λ_n(T^m)}).

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 firstly the simplest case where d = 2.

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 ≥ 0 such that for every l > K the value trace T^l is well defined and trace T^2l+1 = 0 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 N₀, N₀ > M, with the property that T^N₀ ∈ N_2/3(x) (it is possible since a product of three nuclear operators is 2/3-nuclear) and trace T^N₀+2k = 0 for all k = 0, 1, 2, …. By Corollary 3.6, the spectra of all T^N₀+2k are central-symmetric (since, e.g., trace T^N₀ = trace (T^N₀)³ = trace (T^N₀)⁵ = … = 0 by assumption). Assume that the spectrum of T is not central-symmetric. Then there exists an eigenvalue λ₀ ∈ sp (T) such that –λ₀ ∉ sp (T).

Now, λ0N0 ∈ sp (T^N₀), so −λ0N0 ∈ sp (T^N₀). Hence, there exist μ_N₀ ∈ sp (T) and θ_N₀ so that |θ_N₀| = 1, μN0N0=−λ0N0 and μ_N₀ = θ_N₀ λ₀, θ_N₀ ≠ –1. Analogously, λ0N0+2∈sp(TN0+2),so−λ0N0+2∈sp(TN0+2). Hence, there exist μ_N₀+2 ∈ sp (T) and θ_N₀+2 so that |θ_N₀+2| = 1, μN0+2N0+2=−λ0N0+2 and μ_N₀+2 = θ_N₀+2λ₀, θ_N₀+2 ≠ –1 etc. By induction we get the sequences {μN0+2k}k=0∞and{θN0+2k}k=0∞ with the properties that μ_N₀+2k ∈ sp (T), |θ_N₀+2k| = 1, μN0+2kN0+2k=−λ0N0+2k and μ_N₀+2k = θ_N₀+2k λ₀, θ_N₀+2k ≠ –1. Since μ_N₀+2k ∈ sp (T) and |μ_N₀+2k| = |λ₀| > 0, the sequence {μ_N₀+2k is finite as a set, i.e., we have that μ_N₀+2K = μ_N₀+2K+2 = … for some K > 1. It follows that θ_N₀+2K = θ_N₀+2K+2 = …. But θN0+2kN0+2k=−1 for all k. Thus θN0+2Kl=−1 for every odd l ≥ N₀ + 2K. Therefore, θ_N₀+2K = –1. A contradiction. □

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 finite 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 fixed d = 2, 3, … and for the operator T, the spectrum of T is called ℤ_d-symmetric, if λ ∈ sp (T) implies tλ ∈ sp (T) for every t∈1d.

Let r ∈ (0, ∞], 𝔻 := {z ∈ ℂ: |z| < r}, f : 𝔻 → ℂ. and d ∈ ℕ ∖ {1}. We say that f is d-even if f(tz) = f(z) for every t∈1d.

Lemma 5.2

Let Φ(X) be a linear subspace ofX^*⊗̂X of spectral type l₁, i.e., for every v ∈ Φ(X) the series ∑_λ∈sp(v͠) |λ| is convergent. Let d ∈ ℕ, d > 1. If u ∈ Φ(X), then the Fredholm determinant det (1 – zu) is d-even if and only if the eigenvalue sequence of u͠ is ℤ_d-symmetric if and only if trace u^kd+r = 0 for all k = 0, 1, 2, … and r = 1, 2, …, d – 1.

Proof

If the function det (1 – zu) is d-even, then the eigenvalue sequence of u͠ is ℤ_d-symmetric, since this sequence coincides with the sequence of inverses of zeros of det (1 – zu) (according to their multiplicities).

If the eigenvalue sequence of u͠ is ℤ_d-symmetric, then trace u = ∑_{λ∈sp (u͠)} λ = 0 (since Φ(X) is of spectral type l₁ and ∑t∈1dt=0). Also, by the same reason trace u^kd+r = 0 for all k = 0, 1, 2, … and r = 1, 2, …, d – 1, since the spectrum of u͠^l is absolutely summable for every l ≥ 2 and we may assume that {λ_m(u͠^l)} = {λ_m(u͠)^l} (for every fixed l) [12, 3.2.24, p. 147].

Now, let trace u^kd+r = 0 for all k = 0, 1, 2, … and r = 1, 2, …, d – 1. By (1), det (1 – zu) = exp(−∑m=1∞1mdzmdtraceumd) in a neighborhood U of zero. Therefore, this function is d-even in the neighborhood U. By the uniqueness theorem det (1 – zu) is d-even in ℂ. □

Corollary 5.3

For any Banach space X and for every u ∈ X^*⊗̂_2/3X the conclusion of Lemma 5.2 holds.

Corollary 5.4

Let Y be a subspace of a quotient of an L_p-space, 1 ≤ p ≤ ∞. For any u ∈ Y^*⊗̂_sY, where 1/s = 1 + |1/2 – 1/p|, the conclusion of Lemma 5.2 holds.

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 ≥ 0 such that for every l > Kd the value trace T^l is well defined and trace T^kd+r = 0 for all k = K, K + 1, K + 2, … and r = 1, 2, …, d – 1.

Proof

Fix N₀ ∈ ℕ such that T^N₀ is 2/3-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 defined for all l ≥ N₀.

Suppose that the spectrum of T is ℤ_d-symmetric. Take an integer l : = kd + r ≥ N₀ with 0 < 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 = 0.

In proving the converse, we may (and do) assume that Kd > N₀. Consider an infinite increasing sequence {p_m} of prime numbers with p₁ > (K + 1)d. Assuming that trace T^kd+r = 0 if k = K, K + 1, K + 2, … and r = 1, 2, …, d – 1, for a fixed p_m we get from Lemma 5.2 (more precisely, from Corollary 5.3) that the function det (1 – zT^p_m) is d-even. Suppose that the spectrum of T is not ℤ_d-symmetric. Then there exist an eigenvalue λ₀ ∈ sp (T) and a root θ∈1d so that θλ₀ ∉ sp (T). On the other hand, again by Lemma 5.2, the spectrum of T^p_m is ℤ_d-symmetric. Since λ0pm∈sp(Tpm), there exists μ_m ∈ sp (T) such that μmpm=θpmλ0pm∈sp(Tpm); hence, μ_m = θ_m λ₀ for some θ_m with |θ_m| = 1. But |λ₀| > 0. Therefore, the set {μ_m} is finite and it follows that there is an integer M > 1 such that θ_M = θ_M+1 = θ_M+2 = …. Hence, θpm=θMpm 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 ∈ [1, ∞); see [10] for a definition and related facts). Then for some n one hasΠpn⊂N.In particular, Π22(C[0,1])⊂N(C[0,1]),but not every absolutely 2-summing operator in C[0, 1] is compact. Another interesting example: Π33is of spectral type l₁ [16]. We do not know (maybe it is unknown to everybody), whether the finite rank operators are dense in this ideal. However, Theorem 5.5may be applied. Moreover, it can be seen that, for example, the spectrum of an operatorT fromΠ33is central-symmetric if and only if the spectral traces of the operators T^2k–1are zero for all k > 0.

Acknowledgement

I would like to express my gratitude to B. S. Mityagin for a two-days helpful discussion on the topic of this paper and on the problems related to the notion of nuclear operators at Aleksander Pełczyński Memorial Conference (13 July – 19 July, 2014, Będlewo, Poland) as well as for very useful information which B. S. Mityagin gave me in letters. Also, I am very grateful to him for drawing my attention to the paper of M. I. Zelikin [1].

The author would like to thank the referee for helpful remarks.

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Received: 2016-10-31

Accepted: 2018-03-27

Published Online: 2018-04-26

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Articles in the same Issue

https://doi.org/10.1515/math-2018-0043

Keywords for this article

Tensor product; Approximation property; Eigenvalue; Fredholm determinant

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