arXiv:1803.10021v1 [math.FA] 27 Mar 2018 determinant. ftecrepnigtno lmn r zero. are nuclea element the tensor if corresponding only the and nuclear if of a central-symmetric of is spectrum space the same, t Hilbert Ze way: the Thus, following that is the trace. says what in spectral reformulated (or, its [5] space with Lidskiˇı theorem coincides Hilbert element) well-known a tensor on in The o crossnorm operator product greatest nuclear 119]. the tensor any to the respect с. of with completion s [14, dual the Hilbert s Banach a of we its in elements case and operators this the Trace-class in as 77]); also operator). an с. of [14, trace" (see mult operators their trace-class to all according (taken t eigenvalues to its o according all nuclear of any (written of sum eigenvalues trace the spectral all the po and of summable, except system absolutely point This limit non- of no zero. consists have space which Hilbert multiplicity, a zero. algebraic in equal operator operator the nuclear of every central-sy powers of is odd all space of Hilbert traces separable spectral a in operator nuclear e od:nceroeao,tno rdc,approximatio product, tensor operator, nuclear 47B10. words: 47B06, Key 2010: Classification Subject AMS h pc fnceroeaosi ibr pc a edefine be may space Hilbert a in operators nuclear of space The .Introduction 1. oermrso pcr fnceroperators nuclear of spectra on remarks Some l d oeso h prtreulzr.Tecieimcnn Grothendi can from operator criterium follows nuclear It The a exists spaces: zero. there Banach equal general of operator case the the of central-symmetric powers is odd space all Hilbert separable a in operator Abstract. and hr eeaiain fZlknsterm(otecssof cases the (to ar theorem powers of Zelikin’s which of of generalizations some spaces) sharp Banach (in operators compact eesrl compact). necessarily L p U sae)ado iygnsrsl frtecs hr h ope the where case the (for result Mityagin’s of and -spaces) 2 0 = . twssonb .I eii 20)ta h pcrmo nucl a of spectrum the that (2007) Zelikin I. M. by shown was It .Mtai 21)hsgnrlzdZlknscieimt t to criterium Zelikin’s generalized has (2016) Mityagin B. twssonb .I eii n[6 httesetu fa of spectrum the that [16] in Zelikin I. M. by shown was It . U ntespace the in lgI Reinov I. Oleg 1 l 1 ihtepoet that property the with rpry ievle Fredholm eigenvalue, property, n ffteseta rcsof traces spectral the iff usae fquotients of subspaces ii’ hoe 1]cnbe can [16] theorem likin’s mti fadol fthe if only and if mmetric c-norslsthat results eck-Enflo rcso l d powers odd all of traces r ekaotte"nuclear the about peak ealta h spectrum the that Recall eoegnauso finite of eigenvalues zero eao s ydefinition, by is, perator sbyzr,admaybe and zero, ssibly aecnb considered be can pace iplicities). prtri separable a in operator ula.W give We nuclear. e tb xeddto extended be ot ermlilcte)is multiplicities) heir ftecorresponding the of encertaeof trace nuclear he hstno product tensor this h ibr space Hilbert the f aosaenot are rators trace stesaeof space the as d ecs of case he U 1 = ear SOME REMARKS ON SPECTRA OF NUCLEAR OPERATORS 2
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 Lp(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 Zd-symmetry of the spectra of the linear operators. The notion of Zd-symmetry of the spectra was introduced by B. S. Mityagin in a preprint [6] and in his paper [7]. He is interested there in a generalization of the result from [16] 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 Zd-symmetric spectrum. Roughly speaking, Zd-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" num- bers tλ for t √d 1. 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 m T of which is nuclear, is Zd-symmetric if and only if for all large enough integers of type kd + r (0
2. Preliminaries. We denote by X,Y,... 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). h i SOME REMARKS ON SPECTRA OF NUCLEAR OPERATORS 3
By X∗ X we denote the projective tensor product of the spaces X∗ and X ⊗ [2] (see also [11, 12]). It is a completion of the algebraic tensor product X∗ X ⊗ (consideredb as a linear space of all finite rank continuous operators w in X) with respect to the norm N N
w := inf xk′ xk : w = xk′ xk . || ||∧ { || |||| ||! ⊗ } Xk=1 Xk=1 Every element u of the projective tensor product X∗ X can be represented in the form ⊗ u = λix′ xi, b i ⊗ i X where (λ ) l и x′ 1, x 1 [2]. i ∈ 1 || i|| ≤ || i|| ≤ More generally, if 0 < s 1, then X∗ X is a subspace of the projective ≤ ⊗s tensor product, consisting of the tensor elements u,u X∗ X, which admit rep- ∈ ⊗ resentations of the form u = k∞=1 xk′ xkb, where (xk′ ) X∗, (xk) X and s s ⊗ ⊂ ⊂ ∞ x′ xk < [2, 11, 12]. Thus, X∗ X = X∗ 1X. b k=1 || k|| || || ∞ P ⊗ ⊗ On the linear space X∗ X, a linear functional "trace" is defined by a natural P ⊗ way. It is continuous on the normed space b(X∗ X,b ) 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 = ∞ x′ x generates ∈ ⊗ k=1 k ⊗ k naturally an operator u : X X, u(bx) := k∞=1 xk′ , x xk for x X. This defines → h i P ∈ a natural mapping j1 : X∗ X L(bX). The operators, lying in the image of this ⊗ → P map, are called nucleare [2], [8]. Moree generally, if 0 approximation property of order s, 0⊗< s 1 (the AP ), if the canonical ≤ e s mapping js is one-to-one [11, 12]. Note thatb the AP1 is exactly the approximation property AP of A. Grothendieck [2], [8]. Classical spaces, such as Lp(µ) and C(K), have the approximation property. If a space X has the APs, then we can identify the tensor product X∗ X with the space N (X) of all s-nuclear operators in X ⊗s s (i.e. with the image of this tensor product under the map js). In this case for every operator T Ns(X)= bX∗ sX the functional trace T is well defined and called the nuclear trace∈ of the operator⊗ T. It is clear that if a Banachb space has the approximation property, then it has all the properties APs, s (0, 1]. Every Banach space has the property AP2/3 (A. Grothendiek [2], see also [10]).∈ 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 1 [13, 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 some Lp(µ)-space, then it has the APs.
Thus, for such spaces we have an equality Y ∗ sY = Ns(Y ) and the nuclear trace of any operator T N (Y ) is well defined. ⊗ ∈ s b SOME REMARKS ON SPECTRA OF NUCLEAR OPERATORS 4
We will need also the following auxiliary assertion (the first part of which is a consequence of the previous lemma). Lemma 2 [13, Theorem 1]. Let Y be a subspace of a quotient (or a quotient of a subspace) of some Lp(µ)-space, 1 p . If T Ns(Y ), where 1/s = 1+ 1/2 1/p , then ≤ ≤ ∞ ∈ 1.| the− nuclear| trace of the operator T is well defined,
2. n∞=1 λn(T ) < , where λn(T ) is the system of all eigenvalues of the operator T (written| | according∞ to their{ algebraic} multiplicities), and P
∞ trace T = λn(T ). n=1 X
Following [16], we say that a spectrum of a compact operator in a Banach space is central-symmetric, if for every 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 of 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 [16]) can not be extended to the case of general Banach spaces, even if the spaces have the Grothendieck approximation property. Example 1. Let U be a nuclear operator in the space l1, constructed in [8, Proposition 10.4.8]. This operator has the property that trace U = 1 and U 2 = 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 Ns(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 below). Note also that, however, the traces of all operators U m, m = 2, 3,..., (in 2n 1 particular, U − ) are equal to zero. Remark 1. For every nuclear operator T : X X and for any natural number n> 1, the nuclear trace trace T n is well defined (see→ [2, Chap. II, Cor. 2, p. 16]) and equal the sum of all its eigenvalues (according to their multiplicity) [2, 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 of T and m m m T can be arranged in such a way that λn(T ) = λn(T ) (see, e.g., [9, 3.2.24, p. 147]). { } { } Let us formulate and prove the central result of this section. Theorem 1. Let Y be a subspace of a quotient (or a quotient of a subspace) of an Lp-space, 1 p , and u Y ∗ sY, where 1/s =1+ 1/2 1/p , The spectrum ≤ ≤∞ ∈ ⊗ | 2n −1 | of the operator u is central-symmetric if and only if trace u − =0, n =1, 2,.... b e SOME REMARKS ON SPECTRA OF NUCLEAR OPERATORS 5
Proof. If the spectrum of u is central-symmetric, then, by Lemma 2, trace u = m m trace u = n∞=1 λn(T ) = 0; also, by Remark 1, trace u = n∞=1 λn(T )=0 for m =3, 5,.... P e P Toe prove the inverse, 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 [2, Chap. − II, p. 13] or [3, 8, 9]) is an entire function b det (1 zu)=1 z trace u + +( 1)nznα (u)+ ..., − − ··· − n all zeros of which are exactly (according to their multiplicities) the inverses of nonzero eigenvalues of the operator u, associated with the tensor element u. By [2, Chap. II, Cor. 2, pp. 17-18], this entire function is of the form
e ∞ z trace u zzi det (1 zu)= e− (1 zz ) e , − − i i=1 Y where zi = λi(u) is a system of all eigenvalues of the operator u (written according to their{ algebraic multiplicities).} Also, there exists a δ > 0 such that for all z, z δ, | | ≤ we have e e
∞ 1 (1) det (1 zu) = exp ( zn trace un) − − n n=1 X (see [3, p. 350]; cf. [1, Theor. I.3.3, p. 10]). Now, let u Y ∗ sY be as in the formulation of our theorem and suppose that 2n 1 ∈ ⊗ trace u − =0, n =1, 2,.... By (1), we get: for a neighborhood U = U(0) of zero in C, det (1 zu) = det(1+b 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 u. Hence, the spectrum of u is central-symmetric.
Since under the conditions of Theorem 1 the spacee Y has the APs, the tensore product Y ∗ sY can be identified naturally with the space of all s-nuclear operators in Y. Hence,⊗ the statement of Theorem 1 may be reformulated in the following way: b Corollary 1. 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 an Lp-space, T be an s-nuclear operator 2n 1 in Y. The spectrum of T is central-symmetric if and only if trace T − =0, n N. ∈
Corollary 2 [16]. The spectrum of a nuclear operator T, acting on a Hilbert 2n 1 space, is central-symmetric if and only if trace T − =0, n N. For a proof, it is enough to apply Theorem 1 for the case∈p =2. Corollary 3. The spectrum of a 2/3-nuclear operator T, acting on an arbitrary 2n 1 Banach space, is central-symmetric if and only if trace T − =0, n N. ∈ SOME REMARKS ON SPECTRA OF NUCLEAR OPERATORS 6
For a proof, it is enough to apply Theorem 1 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, let us pay attention again to the nuclear operator from Example 1.
4. Sharpness of results of Section 3. Now we will show that the statement of Theorem 1 is sharp and that the exponent s is optimal if p is fixed (if of course p =2, i.e. s = 1). 6 Consider6 the case 2
r, trace wp =1, w˜p =0 and the space Yp (as well as Yp∗) has the APr (but evidently does not have the APs if 1 s>r). Moreover,b this element admitsb a nuclear representation of the form ≥
∞ ∞ s w = µ x′ x , where x′ = x =1, µ < s > r. p k k ⊗ k || k|| || k|| | k| ∞ ∀ Xk=1 Xk=1
Evidently, we have for a tensor element u := wp from the assertion (⋆) : trace u = 1 and the spectrum of the operator u equals 0 . The case where 2
Proof. Suppose that p > 2. Consider the tensor element w := wp from the assertion (⋆) and its representation w = k∞=1 µk xk′ xk, where xk′ = xk = 1 s ⊗ || || || || и ∞ µk < for each s, s > r. Let l : Y := Yp lp be the identity inclusion. k=1 | | ∞ P → Let y′ be an extension of the functional x′ (k = 1, 2,... ) from the subspace Y P k k to the whole space l with the same norm and set v := ∞ µ y′ l(x ). Then p k=1 k k ⊗ k v l ′ l (1/p +1/p′ = 1) for each s (r, 1], trace v = µ y′ ,l(x ) =1 and ∈ p ⊗s p ∈ P k h k k i v(lp) l(Y ) lp. On the other hand, we have a diagram: ⊂ ⊂ P b e e e Y l l v0 Y l l v0 Y v0 l , e → p → → p → → p SOME REMARKS ON SPECTRA OF NUCLEAR OPERATORS 7 where v0 is an operator generated by v, v = lv0 and v0l = w = 0. Put V := v. 2 Clearly, trace V = 1 and the spectrum sp V = 0 . Let us note that V / Nr(lp) (by Lemma 2). If p [1, 2), then it is enough to consider{ } the adjoint operator.∈ e ∈ e e e e e e It follows from Theorem 2 that the assertion of Corollary 1 is optimal already in the case of the space Y = lp (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., [4, p. 943] for a definition),∈ then T is a Riesz∈ operator too (see, e.g., [9, 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 m m λn(T ) = λn(T ) (see [9, 3.2.24, p. 147]). Recall that in this case we denote { } { } m m m by sp (T ) (resp., by sp (T ) the sequence λn(T ) (resp., λn(T ) ). We are going to present a short proof of{ the theorem} of{ B. Mitya}gin from [6, 7]. To clarify our idea of the proof, let us consider firstly the simplest case where d =2.
Theorem 3. 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 N , N > M, ∈with the property that T N0 ∈ N (X) (it is∈ possible 0 0 ∈ 2/3 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, 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 ). N0 ∈ N0 N−0 ∈ N0 Now, λ0 sp (T ), so λ0 sp (T ). Hence, there exist µN0 sp (T ) and ∈ N0 − ∈N0 ∈ θN0 so that θN0 = 1, µN0 = λ0 and µN0 = θN0 λ0, θN0 = 1. Analogously, N0+2 | N0+2| N0+2 − N0+2 6 − λ0 sp (T ), so λ0 sp (T ). Hence, there exist µN0+2 sp (T ) and ∈ − N0+2∈ N0+2 ∈ θ so that θ =1, µ = λ and µ = θ λ , θ = 1 etc. N0+2 | N0+2| N0+2 − 0 N0+2 N0+2 0 N0+2 6 − By induction we get the sequences µN0+2k k∞=0 and θN0+2k k∞=0 with the properties { N0+2}k N{0+2k } that µN0+2k sp (T ), θN0+2k = 1, µN0+2k = λ0 and µN0+2k = θN0+2kλ0, θ = 1.∈Since µ | sp| (T ) and µ −= λ > 0, the sequence µ N0+2k 6 − N0+2k ∈ | N0+2k| | 0| { N0+2k} is finite as a set, i.e., we have that µN0+2K = µN0+2K+2 = ... for some K > 1. It N0+2k l follows that θN0+2K = θN0+2K+2 = .... But θN0+2k = 1 for all k. Thus θN0+2K = 1 for every odd l N +2K. Therefore, θ = 1.−Contradiction. − ≥ 0 N0+2K − Now we are going to consider a general case of a notion of Zd-symmetry of a spectra, introduced and investigated by B. Mityagin in [6, 7]. Let T be an operator in X, all non-zero spectral values of which are eigenvalues of finite multiplicity and have no limit point except possibly zero. Recall that we denote by sp (T ) the SOME REMARKS ON SPECTRA OF NUCLEAR OPERATORS 8 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 Zd-symmetric, if λ sp (T ) implies tλ sp (T ) for every t √d 1. Let∈ r (0, ], D :=∈ z C : z < r ∈, f : D C. and d N 1 . We say ∈ ∞ { ∈ | | } d → ∈ \{ } that f is d-even if f(tz)= f(z) for every t √1. ∈ Lemma 3. Let Φ(X) be a linear subspace of X∗ X of spectral type l1, i.e., for ⊗ N every v Φ(X) the series λ sp (ve) λ is convergent. Let d ,d> 1. If u Φ(X), ∈ ∈ | | ∈ ∈ then the Fredholm determinant det (1 zu) is d-evenb if and only if the eigenvalue P − kd+r sequence of u is Zd-symmetric if and only if trace u = 0 for all k = 0, 1, 2,... and r =1, 2,...,d 1. − Proof. Ife the function det (1 zu) is d-even, then the eigenvalue sequence of u − is Zd-symmetric, since this sequence coincides with the sequence of inverses of zeros of det (1 zu) (according to their multiplicities). e − Z If the eigenvalue sequence of u is d-symmetric, then trace u = λ sp (ue) λ = 0 ∈ d (since Φ(X) is of spectral type l1 and t √1 t = 0). Also, by the same reason ∈ P trace ukd+r =0 for all k =0, 1, 2,...e and r =1, 2,...,d 1, since the spectrum of ul P is absolutely summable for every l 2 and we may assume− that λ (ul) = λ (u)l ≥ { m } { m } (for every fixed l) [9, 3.2.24, p. 147]. e kd+r Now, let trace u = 0 for all k = 0, 1, 2,... and r = 1, 2e,...,d 1.eBy 1 md md − ∞ (1), det (1 zu) = exp ( m=1 md z trace u ) 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. P − Corollary 4. For any Banach space X and for every u X∗ 2/3X the conclusion of Lemma 3 holds. ∈ ⊗ Corollary 5. Let Y be a subspace of a quotient of an L -space,b 1 p . For p ≤ ≤∞ any u Y ∗ Y, where 1/s =1+ 1/2 1/p , the conclusion of Lemma 3 holds. ∈ ⊗s | − | Now we are ready to present a short proof of the theorem of B. Mityagin [6, 7]. Note that theb theorem in [6, 7] is formulated and proved for compact operators, but the proof from [6, 7] can be easily adapted for the general case of linear operators. Theorem 4. Let X be a Banach space and T L(X). Suppose that some ∈ power of T is nuclear. The spectrum of T is Zd-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. N0 − Proof. Fix N0 N such that T is 2/3-nuclear (it is possible by a composition theorem from [2, Chap,∈ II, Theor. 3, p. 10]). Note that, by A. Grothendieck, the trace of T l is well defined for all l N . ≥ 0 Suppose that the spectrum of T is Zd-symmetric. Take an integer l := kd + r l l ≥ N0 with 0 < r < d. Since the spectrum of T is absolutely summable, trace T = l l kd+r λ sp (T l) λ and we may assume that λm(T ) = λm(T ) , we get that trace T = 0. ∈ { } { } P In proving the converse, we may (and do) assume that Kd> N0. Consider an infinite increasing sequence pm 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 − SOME REMARKS ON SPECTRA OF NUCLEAR OPERATORS 9
fixed pm we get from Lemma 3 (more precisely, from Corollary 4) that the function pm det (1 zT ) is d-even. Suppose that the spectrum of T is not Zd-symmetric. Then − d there exist an eigenvalue λ0 sp (T ) and a root θ √1 so that θλ0 / sp (T ). On ∈ ∈ pm ∈ the other hand, again by Lemma 3, the spectrum of T is Zd-symmetric. Since pm pm pm pm pm pm λ0 sp (T ), there exists µm sp (T ) such that µm = θ λ0 sp (T ); hence, µ =∈θ λ for some θ with θ ∈=1. But λ > 0. Therefore, the∈ set µ is finite m m 0 m | m| | 0| { m} and it follows that there is an integer M > 1 such that θM = θM+1 = θM+2 = .... Hence, θpm = θpm for all m M. Thus, θ = θ. Contradiction. M ≥ M Let us give some examples in which we can apply Theorem 4, but the main result of [6, 7] does not work.
Example 2. Let Πp be the ideal of absolutely p-summing operators (p [1, ); n ∈ ∞ see [8] for a definition and related facts). Then for some n one has Πp N. In 2 ⊂ particular, Π2(C[0, 1]) N(C[0, 1]), but not every absolutely 2-summing operator ⊂ 3 in C[0, 1] is compact. Another interesting example: Π3 is of spectral type l1 [15]. We do not know (maybe, it is not known to everybody), whether the finite rank operators are dense in this ideal. However, Theorem 4 may be applied. Moreover, 3 it can be seen that, for example, the spectrum of an operator T from Π3 is central- 2k 1 symmetric if and only if the spectral traces of the operators T − are 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 Pel czy´nski Memorial Conference (13 July – 19 July, 2014, B¸edlewo, 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 [16].
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[11] Reinov O. I., Approximation properties APs and p-nuclear operators (the case 0
Saint Petersburg State University E-mail address: [email protected]