Stability of versions of the Weyl-type theorems under tensor product
We study the transformation versions of the Weyl-type theorems from operators $T$ and $S$ for their tensor product $T \otimes S$ in the infinite-dimensional space setting.
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Ukrains’kyi Matematychnyi Zhurnal| _version_ | 1860507738909442048 |
|---|---|
| author | Prasad, T. Rashid, M. H. M. Прасад, Т. Рашид, М. Х. М. |
| author_facet | Prasad, T. Rashid, M. H. M. Прасад, Т. Рашид, М. Х. М. |
| author_sort | Prasad, T. |
| baseUrl_str | https://umj.imath.kiev.ua/index.php/umj/oai |
| collection | OJS |
| datestamp_date | 2019-12-05T09:29:54Z |
| description | We study the transformation versions of the Weyl-type theorems from operators $T$ and $S$ for their tensor product $T \otimes S$ in the infinite-dimensional space setting. |
| first_indexed | 2026-03-24T02:14:05Z |
| format | Article |
| fulltext |
UDC 515.14
M. H. M. Rashid, T. Prasad (Mu’tah Univ., Al-Karak, Jordan)
STABILITY OF VERSIONS OF THE WEYL-TYPE THEOREMS
UNDER TENSOR PRODUCT
СТАБIЛЬНIСТЬ РIЗНИХ ВЕРСIЙ ТЕОРЕМ ТИПУ ВЕЙЛЯ
ДЛЯ ТЕНЗОРНОГО ДОБУТКУ
We study the transformation versions of the Weyl-type theorems from operators T and S for their tensor product T \otimes S in
the infinite-dimensional space setting.
Вивчаються трансформованi версiї теорем типу Вейля для операторiв T i S та їх тензорного добутку T \otimes S у
нескiнченновимiрнiй постановцi.
1. Introduction. Given Banach spaces \scrX and \scrY , let \scrX \otimes \scrY denote the completion (in some reasonable
uniform cross norm) of the tensor product of \scrX and \scrY . For Banach space operators A \in \scrB (\scrX ) and
B \in \scrB (\scrY ), let A\otimes B \in \scrB (\scrX \otimes \scrY ) denote the tensor product of A and B. Recall that for an operator
S, the Browder spectrum \sigma b(S) and the Weyl spectrum \sigma w(S) of S are the sets
\sigma b(S) = \{ \lambda \in \BbbC : S - \lambda is not Fredholm or \mathrm{a}\mathrm{s}\mathrm{c} (S - \lambda ) \not = \mathrm{d}\mathrm{s}\mathrm{c} (S - \lambda )\} ,
\sigma w(S) = \{ \lambda \in \BbbC : S - \lambda is not Fredholm or \mathrm{i}\mathrm{n}\mathrm{d} (S - \lambda ) \not = 0\} .
In the case in which \scrX and \scrY are Hilbert spaces, Kubrusly and Duggal [13] proved that
\mathrm{i}\mathrm{f} \sigma b(A) = \sigma w(A) \mathrm{a}\mathrm{n}\mathrm{d} \sigma b(B) = \sigma w(B), \mathrm{t}\mathrm{h}\mathrm{e}\mathrm{n} \sigma b(A\otimes B) = \sigma w(A\otimes B)
\mathrm{i}\mathrm{f} \mathrm{a}\mathrm{n}\mathrm{d} \mathrm{o}\mathrm{n}\mathrm{l}\mathrm{y} \mathrm{i}\mathrm{f} \sigma w(A\otimes B) = \sigma (A)\sigma w(B) \cup \sigma w(A)\sigma (B).
In other words, if A and B satisfy Browder’s theorem, then their tensor product satisfies Brow-
der’s theorem if and only if the Weyl spectrum identity holds true. The same proof still holds in a
Banach space setting. Recently, Rashid and Prasad studied property (Sw): a Banach space operator T,
T \in \scrB (\scrX ), satisfies property (Sw) if \sigma (T )\setminus \sigma SBF -
+
(T ) = E0(T ), where \sigma denote the usual spectrum,
\sigma SBF -
+
(T ) = \{ \lambda \in \BbbC : T - \lambda is not an upper B-Fredholm or \mathrm{i}\mathrm{n}\mathrm{d} (T - \lambda ) > 0\} denotes the upper
B-Weyl spectrum and E0(T ) = \{ \lambda \in \mathrm{i}\mathrm{s}\mathrm{o}\sigma (T ) : 0 < \alpha (T - \lambda ) < \infty \} is the set of finite multiplicity
isolated eigenvalues of T and that T \in \scrB (\scrX ) obeys property (Sb) if \sigma (T ) \setminus \sigma SBF -
+
(T ) = \pi 0(T ),
where \pi 0(T ) is the set of all poles of finite rank. This paper intents to discuss the stability of property
(Sb) and property (Sw) under tensor product T \otimes S of Banach space operators T and S.
2. Notation and complementary results. For a bounded linear operator S \in \scrB (\scrX ), let \sigma (S),
\sigma p(S) and \sigma a(S) denote, respectively, the spectrum, the point spectrum and the approximate point
spectrum of S and if G \subseteq \BbbC , then \mathrm{i}\mathrm{s}\mathrm{o}G denote the isolated points of G. Let \alpha (S) and \beta (S)
denote the nullity and the deficiency of S, defined by \alpha (S) = \mathrm{d}\mathrm{i}\mathrm{m}\mathrm{k}\mathrm{e}\mathrm{r}(S) and \beta (S) = \mathrm{c}\mathrm{o}\mathrm{d}\mathrm{i}\mathrm{m}\Re (S).
If the range \Re (S) of S is closed and \alpha (S) < \infty (resp. \beta (S) < \infty ), then S is called an upper
semi-Fredholm (resp. a lower semi-Fredholm) operator. If S \in \scrB (\scrX ) is either upper or lower semi-
Fredholm, then S is called a semi-Fredholm operator, and \mathrm{i}\mathrm{n}\mathrm{d} (S), the index of S, is then defined
by \mathrm{i}\mathrm{n}\mathrm{d} (S) = \alpha (S) - \beta (S). If both \alpha (S) and \beta (S) are finite, then S is a Fredholm operator. The
c\bigcirc M. H. M. RASHID, T. PRASAD, 2016
542 ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 4
STABILITY OF VERSIONS OF THE WEYL-TYPE THEOREMS UNDER TENSOR PRODUCT 543
ascent, denoted \mathrm{a}\mathrm{s}\mathrm{c} (S), and the descent, denoted \mathrm{d}\mathrm{s}\mathrm{c} (S), of S are given by \mathrm{a}\mathrm{s}\mathrm{c} (S) = \mathrm{i}\mathrm{n}\mathrm{f}\{ n \in \BbbN :
\mathrm{k}\mathrm{e}\mathrm{r} (Sn) = \mathrm{k}\mathrm{e}\mathrm{r} (Sn+1)\} , \mathrm{d}\mathrm{s}\mathrm{c} (S) = \mathrm{i}\mathrm{n}\mathrm{f}\{ n \in \BbbN : \Re (Sn) = \Re (Sn+1)\} (where the infimum is taken
over the set of nonnegative integers; if no such integer n exists, then \mathrm{a}\mathrm{s}\mathrm{c} (S) = \infty , respectively
\mathrm{d}\mathrm{s}\mathrm{c} (S) = \infty ).
According to Coburn [7], Weyl’s theorem holds for S if \Delta (S) = \sigma (S) \setminus \sigma w(S) = E0(S), and
that Browder’s theorem holds for S(S \in \scrB ) if \Delta (S) = \sigma (S) \setminus \sigma w(S) = \pi 0(S), or equivalently
\sigma b(S) = \sigma w(T ).
For S \in \scrB (\scrX ) and a nonnegative integer n define S[n] to be the restriction of S to \Re (Sn)
viewed as a map from \Re (Sn) into \Re (Sn) (in particular, S[0] = S). If for some integer n the
range space \Re (Sn) is closed and S[n] is an upper (a lower) semi-Fredholm operator, then S is
called an upper (a lower) semi-B-Fredholm operator. In this case the index of S is defined as the
index of the semi-B-Fredholm operator S[n], see [4]. Moreover, if S[n] is a Fredholm operator,
then S is called a B-Fredholm operator. A semi-B-Fredholm operator is an upper or a lower
semi-B-Fredholm operator. An operator S is said to be a B-Weyl operator [3] (Definition 1.1)
if it is a B-Fredholm operator of index zero. The B-Weyl spectrum \sigma BW (S) of S is defined by
\sigma BW (S) = \{ \lambda \in \BbbC : S - \lambda I \mathrm{i}\mathrm{s} \mathrm{n}\mathrm{o}\mathrm{t} \mathrm{a} \mathrm{B} - \mathrm{W}\mathrm{e}\mathrm{y}\mathrm{l} \mathrm{o}\mathrm{p}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{o}\mathrm{r}\} . An operator S \in \scrB (\scrX ) is called Drazin
invertible if it has a finite ascent and descent. The Drazin spectrum \sigma D(S) of an operator S is
defined by \sigma D(S) = \{ \lambda \in \BbbC : S - \lambda I is not Drazin invertible\} . Define also the set LD(\scrX ) by
LD(\scrX ) = \{ S \in \scrB (\scrX ) : a(S) < \infty \mathrm{a}\mathrm{n}\mathrm{d} \Re (Sa(S)+1) \mathrm{i}\mathrm{s} \mathrm{c}\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{e}\mathrm{d}\} and
\sigma LD(S) = \{ \lambda \in \BbbC : S - \lambda /\in LD(\scrX )\} .
Following [2], an operator S \in \scrB (\scrX ) is said to be left Drazin invertible if S \in LD(\scrX ). We say that
\lambda \in \sigma a(S) is a left pole of S if S - \lambda I \in LD(X), and that \lambda \in \sigma a(S) is a left pole of S of finite
rank if \lambda is a left pole of S and \alpha (S - \lambda I) < \infty . Let \pi a(S) denotes the set of all left poles of S and
let \pi 0
a(S) denotes the set of all left poles of S of finite rank. From [2] (Theorem 2.8) it follows that if
S \in \scrB (\scrX ) is left Drazin invertible, then S is an upper semi-B-Fredholm operator of index less than
or equal to 0. Note that \pi a(S) = \sigma (S) \setminus \sigma LD(S) and hence \lambda \in \pi a(S) if and only if \lambda /\in \sigma LD(S).
According to [17], T \in \scrB (\scrX ) satisfies property (Bw) if \sigma (T ) \setminus \sigma BW (T ) = E0(T ). We say that
T satisfies property (Bb) if \sigma (T ) \setminus \sigma BW (T ) = \pi 0(T ) [18]. Property (Bw) implies Weyl’s theorem
but converse is not true also property (Bw) implies property (Bb) but converse is not true [18]. Let
\scrS \scrB \scrF -
+(\scrX ) denote the class of all upper B-Fredholm operators such that ind(T ) \leq 0. The upper
B-Weyl spectrum \sigma SBF -
+
(T ) of T is defined by \sigma SBF -
+
(T ) = \{ \lambda \in \BbbC : T - \lambda /\in \scrS \scrB \scrF -
+(\scrX )\} .
Rashid and Prasad [20] introduced and studied new versions of the Weyl-type theorems property
(Sw) and property (Sb).
Definition 2.1. A bounded linear operator T \in \scrB (\scrX ) is said to satisfy
(i) property (Sw) if \sigma (T ) \setminus \sigma SBF -
+
(T ) = E0(T ) [20],
(ii) property (Sb) if \sigma (T ) \setminus \sigma SBF -
+
(T ) = \pi 0(T ) [20],
(iii) property(Bgw) if \sigma a(T ) \setminus \sigma SBF -
+
(T ) = E0(T ) [18].
The operator T \in \scrB (\scrX ) is said to have the single valued extension property at \lambda 0 \in \BbbC (abbreviated
SVEP at \lambda 0) if for every open disc \BbbD centred at \lambda 0, the only analytic function f : \BbbD \rightarrow \scrX which
satisfies the equation (T - \lambda )f(\lambda ) = 0 for all \lambda \in \BbbD is the function f \equiv 0. An operator T \in \scrB (\scrX )
is said to have SVEP if T has SVEP at every point \lambda \in \BbbC . Obviously, every T \in \scrB (\scrX ) has SVEP
at the points of the resolvent \rho (T ) := \BbbC \setminus \sigma (T ). Moreover, from the identity theorem for analytic
function, it easily follows that T \in \scrB (\scrX ), as well as its dual T \ast , has SVEP at every point of the
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 4
544 M. H. M. RASHID, T. PRASAD
boundary \partial \sigma (T ) = \partial \sigma (T \ast ) of the spectrum \sigma (T ). In particular, both T and T \ast have SVEP at every
isolated point of the spectrum, see [1]. Let T \in \scrB (\scrX ) and let s \in \BbbN then T has uniform descent for
n \geq s if \Re (T ) + \mathrm{k}\mathrm{e}\mathrm{r}(Tn) = \Re (T ) + \mathrm{k}\mathrm{e}\mathrm{r}(T s) for all n \geq s. If in addition \Re (T ) + \mathrm{k}\mathrm{e}\mathrm{r}(T s) is closed,
then T is said to have topological descent for n \geq s [10]. Let
\scrS \scrF +(S) = \{ \lambda \in \BbbC : S - \lambda \mathrm{i}\mathrm{s} \mathrm{u}\mathrm{p}\mathrm{p}\mathrm{e}\mathrm{r} \mathrm{s}\mathrm{e}\mathrm{m}\mathrm{i}-\mathrm{F}\mathrm{r}\mathrm{e}\mathrm{d}\mathrm{h}\mathrm{o}\mathrm{l}\mathrm{m}\} ,
\scrF (S) = \{ \lambda \in \BbbC : S - \lambda \mathrm{i}\mathrm{s} \mathrm{F}\mathrm{r}\mathrm{e}\mathrm{d}\mathrm{h}\mathrm{o}\mathrm{l}\mathrm{m}\} ,
\sigma SF+(S) = \{ \lambda \in \sigma a(S) : \lambda /\in \scrS \scrF +(S)\} ,
\sigma SF -
+
(S) = \{ \lambda \in \sigma a(S) : \lambda \in \sigma SF+(S) \mathrm{o}\mathrm{r} \mathrm{i}\mathrm{n}\mathrm{d} (S - \lambda ) > 0\} ,
\sigma ub(S) = \{ \lambda \in \sigma a(S) : \lambda \in \sigma SF+(S) \mathrm{o}\mathrm{r} \mathrm{a}\mathrm{s}\mathrm{c} (S - \lambda ) = \infty \} ,
\pi 0
a(S) = \{ \lambda \in \mathrm{i}\mathrm{s}\mathrm{o}\sigma a(S) : \lambda \in \scrS \scrF +(S), \mathrm{a}\mathrm{s}\mathrm{c} (S - \lambda ) < \infty \} ,
E0
a(S) = \{ \lambda \in \mathrm{i}\mathrm{s}\mathrm{o}\sigma a(S) : 0 < \alpha (S - \lambda ) < \infty \} ,
\scrS \scrB \scrF +(S) = \{ \lambda \in \BbbC : S - \lambda \mathrm{i}\mathrm{s} \mathrm{u}\mathrm{p}\mathrm{p}\mathrm{e}\mathrm{r} \mathrm{s}\mathrm{e}\mathrm{m}\mathrm{i}-\itB -\mathrm{F}\mathrm{r}\mathrm{e}\mathrm{d}\mathrm{h}\mathrm{o}\mathrm{l}\mathrm{m}\} ,
\scrS \scrB \scrF (S) = \{ \lambda \in \BbbC : S - \lambda \mathrm{i}\mathrm{s} \itB -\mathrm{F}\mathrm{r}\mathrm{e}\mathrm{d}\mathrm{h}\mathrm{o}\mathrm{l}\mathrm{m}\} ,
\sigma SBF+(S) = \{ \lambda \in \sigma a(S) : \lambda /\in \scrS \scrB \scrF +(S)\} ,
\sigma SBF -
+
(S) = \{ \lambda \in \sigma a(S) : \lambda \in \sigma SBF+(S) \mathrm{o}\mathrm{r} \mathrm{i}\mathrm{n}\mathrm{d} (S - \lambda ) > 0\} ,
H0(S) = \{ x \in \scrX : \mathrm{l}\mathrm{i}\mathrm{m}
n\rightarrow \infty
\| Snx\| 1/n = 0\} ,
\Delta g(S) = \{ \lambda \in \BbbC : \lambda \in \sigma (S) \setminus \sigma BW (S)\} ,
\Delta g
a(S) = \{ \lambda \in \BbbC : \lambda \in \sigma a(S) \setminus \sigma SBF -
+
(S)\} .
Recall that \sigma SF -
+
(S) is the Weyl approximate point spectrum of S, \sigma ub(S) is the Browder approximate
point spectrum of S, and H0(S) is the quasinilpotent of S [1].
We say that S \in \scrB (\scrX ) satisfies a-Browder’s theorem (S \in a\scrB ) if \sigma SF -
+
(S) = \sigma ub(S) or
equivalently, \Delta a(S) = \sigma a(S) \setminus \sigma SF -
+
(S) = \pi 0
a(S) and that S \in \scrB (\scrX ) satisfies a-Weyl’s theorem
(S \in a\scrW ) if \Delta a(S) = Ea
0 (S) [21].
Lemma 2.1. Let T \in \scrB (\scrX ) and S \in \scrB (\scrY ). Then
(i) \sigma x(T \otimes S) = \sigma x(T )\sigma x(S), where \sigma x = \sigma or \sigma a [5, 22],
(ii) \sigma SF+(T \otimes S) = \sigma SF+(T )\sigma a(S) \cup \sigma a(T )\sigma SF+(S) [8].
Recall that an operator T is said to be isoloid if \lambda \in \mathrm{i}\mathrm{s}\mathrm{o}\sigma (T ) implies \lambda \in \sigma p(T ) and that
T \in \scrB (\scrX ) is said to be a-isoloid if \lambda \in \mathrm{i}\mathrm{s}\mathrm{o}\sigma a(T ) implies \lambda \in \sigma p(T ). It is well-known that if T is
a-isoloid, then T is isoloid but not conversely.
Lemma 2.2. Let T \in \scrB (\scrX ) and S \in \scrB (\scrY ). If T and S are isoloid, then
(i) T \otimes S is isoloid [11],
(ii) E0(T \otimes S) \subseteq E0(T )E0(S) [14].
Lemma 2.3 ([9], Theorem 3). If T and S satisfy Browder’s theorem, then the following condi-
tions are equivalent:
(i) T \otimes S \in \scrB ,
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 4
STABILITY OF VERSIONS OF THE WEYL-TYPE THEOREMS UNDER TENSOR PRODUCT 545
(ii) \sigma w(T \otimes S) = \sigma (T )\sigma w(S) \cup \sigma w(T )\sigma (S),
(iii) T has SVEP at points \mu \in \scrF (T ) and S has SVEP at points \nu \in \scrF (S) such that (0 \not =)\lambda =
= \mu \nu /\in \sigma w(T \otimes S).
3. Property (\bfitS \bfitw ) and tensor product. We first give some useful lemmas.
Lemma 3.1. Let T \in \scrB (\scrX ). If T obeys property (Sb) or satisfy any one of the following two
conditions:
(i) \sigma SBF -
+
(T ) = \sigma b(T ),
(ii) \sigma SBF -
+
(T ) \cup E0(T ) = \sigma (T ).
Then the following statements are equivalent:
(i) T obeys property (Sw),
(ii) \sigma SBF -
+
(T ) \cap E0(T ) = \varnothing ,
(iii) E0(T ) = \pi 0(T ).
Let H0(T ) = \{ x \in \scrX : \mathrm{l}\mathrm{i}\mathrm{m}n\rightarrow \infty \| Tnx\|
1
n = 0\} and K(T )=\{ x \in \scrX : there exists a sequence
\{ xn\} \subset \scrX and \delta > 0 for which x = x0, T (xn+1) = xn and \| xn\| \leq \delta n\| x\| for all n = 1, 2, . . .\}
denotes the quasinilpotent part and the analytic core of T \in \scrB (\scrX ). It is well known that H0(T ) and
K(T ) are nonclosed hyperinvariant subspace of \scrX such that T - q(0) \subseteq H0(T ) for all q = 0, 1, 2, . . .
and TK(T ) = K(T ) [15].
Lemma 3.2. Let T \in \scrB (\scrX ) and S \in \scrB (\scrY ) obey property (Sb). Then T \otimes S obeys property (Sb)
if and only if \sigma SBF -
+
(T \otimes S) = \sigma SBF -
+
(T )\sigma (S) \cup \sigma SBF -
+
(S)\sigma (T ).
Proof. First, we have to show that \sigma SBF -
+
(T \otimes S) \subseteq \sigma SBF -
+
(T )\sigma (S) \cup \sigma SBF -
+
(S)\sigma (T ). Let
\lambda /\in \sigma SBF -
+
(T )\sigma (S) \cup \sigma SBF -
+
(S)\sigma (T ). For every factorization \lambda = \mu \nu such that \mu \in \sigma (T ) and
\nu \in \sigma (S) we have that \mu \in \sigma (T )\setminus \sigma SBF -
+
(T ) and \mu \in \sigma (S)\setminus \sigma SBF -
+
(S). That is, T - \mu I and S - \nu I
are upper semi-B- Fredholm operators. In particular \lambda /\in \sigma SBF+(T \otimes S). Now we obtain to prove that
ind(T\otimes S - \lambda ) \leq 0. If \mathrm{i}\mathrm{n}\mathrm{d} (T\otimes S - \lambda ) > 0, then it follows that (T\otimes S - \lambda ) \leq \infty have finite indices and
so (T\otimes S - \lambda ) \in \scrF . Let E = \{ (\mu i, \nu i) \in \sigma (T )\sigma (S) : 1 \leq i \leq p, \mu i\nu i = \lambda \} . Then from [12] (Theorem
3.5) \mathrm{i}\mathrm{n}\mathrm{d}(T \otimes S - \lambda ) =
\sum p
j=n+1
\mathrm{i}\mathrm{n}\mathrm{d} (T - \mu j)\mathrm{d}\mathrm{i}\mathrm{m}H0(S - \nu j)+
\sum n
j=1
\mathrm{i}\mathrm{n}\mathrm{d} (S - \nu j)\mathrm{d}\mathrm{i}\mathrm{m}H0(T - \nu j).
Since \mathrm{i}\mathrm{n}\mathrm{d} (T - \mu i) < 0 and \mathrm{i}\mathrm{n}\mathrm{d} (S - \nu i) < 0, we get a contradiction. Consequently, \lambda /\in \sigma SBF -
+
(T \otimes
\otimes S). Since the inclusion \sigma w(T )\sigma (S)\cup \sigma w(S)\sigma (T ) \subseteq \sigma b(T )\sigma (S)\cup \sigma b(S)\sigma (T ) = \sigma b(T\otimes S) is true and
since \sigma SBF -
+
(T ) \subseteq \sigma w(T ) and \sigma SBF -
+
(S) \subseteq \sigma w(S), we have \sigma SBF -
+
(T \otimes S) \subseteq \sigma SBF -
+
(T )\sigma (S) \cup
\cup \sigma SBF -
+
(S)\sigma (T ) \subseteq \sigma w(T )\sigma (S) \cup \sigma w(S)\sigma (T ) \subseteq \sigma b(T )\sigma (S) \cup \sigma b(S)\sigma (T ) = \sigma b(T \otimes S). Then the
equality \sigma SBF -
+
(T \otimes S) = \sigma SBF -
+
(T )\sigma (S)\cup \sigma SBF -
+
(S)\sigma (T ) follows from Lemma 3.1. Conversely,
suppose the equality \sigma SBF -
+
(T \otimes S) = \sigma SBF -
+
(T )\sigma (S) \cup \sigma SBF -
+
(S)\sigma (T ) holds. Since T and S
satisfy property (Sb), it follows that
\sigma SBF -
+
(T \otimes S) = \sigma SBF -
+
(T )\sigma (S) \cup \sigma SBF -
+
(S)\sigma (T ) =
= \sigma b(T )\sigma (S) \cup \sigma b(S)\sigma (T ) = \sigma b(T \otimes S).
That is, T \otimes S obeys property (Sb).
Lemma 3.2 is proved.
In [14], Kubrusly and Duggal studied the stability of Weyl’s theorem under tensor product in the
infinite dimensional space setting. Rashid [19] studied the stability of generalized Weyl’s theorem
under tensor product in the infinite dimensional Banach space. The following main theorem shows if
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 4
546 M. H. M. RASHID, T. PRASAD
isoloid operators T and S satisfies property (Sw) and the equality \sigma SBF -
+
(T\otimes S) = \sigma SBF -
+
(T )\sigma (S)\cup
\cup \sigma SBF -
+
(S)\sigma (T ) holds, then T \otimes S satisfies property (Sw) in the infinite dimensional space setting.
Let \sigma PF (T ) = \{ \lambda \in \sigma p(T ) : \alpha (T - \lambda ) < \infty \} = \{ \lambda \in \BbbC : 0 < \alpha (T - \lambda ) < \infty \} .
Theorem 3.1. If isoloid operators T and S satisfies property (Sw) and the equality \sigma SBF -
+
(T \otimes
\otimes S) = \sigma SBF -
+
(T )\sigma (S) \cup \sigma SBF -
+
(S)\sigma (T ) holds, then T \otimes S satisfies property (Sw).
Proof. Since T and S satisfies property (Sw), T and S satisfies property (Sb) by [20] (Theorem
2.7). Then by the equality hypothesis \sigma SBF -
+
(T \otimes S) = \sigma SBF -
+
(T )\sigma (S) \cup \sigma SBF -
+
(S)\sigma (T ), T \otimes S
satisfies property (Sb) (see Lemma 3.2). Suppose T \otimes S does not satisfies property (Sw). Then we
have the result \sigma SBF -
+
(T \otimes S) \cap E0(T \otimes S) \not = \varnothing .
Since \sigma SBF -
+
(T \otimes S) = \sigma SBF -
+
(T )\sigma (S) \cup \sigma SBF -
+
(S)\sigma (T ), we get \lambda = \mu \upsilon \in \sigma SBF -
+
(T \otimes S)
if and only if (\mu , \upsilon ) \in \sigma SBF -
+
(T )\sigma (S) or (\mu , \upsilon ) \in \sigma SBF -
+
(S)\sigma (T ). If \lambda \in E0(T \otimes S), then by
applying [14] (Lemma 3), \lambda \in E0(T )E0(S). Thus if, \lambda = \mu \upsilon \in \sigma SBF -
+
(T \otimes S)\cap E0(T \otimes S), then it
follows that 0 \not = \lambda = \mu \nu = \mu \prime \nu \prime with \mu =
\lambda
\nu
\in \sigma SBF -
+
(T ), \mu \prime =
\lambda
\nu \prime
\in E0(S), \nu =
\lambda
\mu
\in \sigma SBF -
+
(S),
\nu \prime =
\lambda
\nu \prime
\in E0(T ). Thus, E0(T ) \not = \varnothing and E0(S) \not = \varnothing . Since \lambda = \mu \nu \in E0(T \otimes S), it follows
by [14] (Lemma 5) that \mu \in \sigma iso(T ) and \nu \in \sigma iso(S). Since T and S are isoloid, and since
\lambda = \mu \nu \in E0(T \otimes S), it follows that \mu \in \sigma PF (T ) and \nu \in \sigma PF (S). Since T and S satisfies property
(Sw), \mu \in \sigma SBF -
+
(T ) \cap \sigma iso(T ) \cap \sigma PF (T ) and \nu \in \sigma SBF -
+
(S) \cap \sigma iso(S) \cap \sigma PF (S) which implies
that both \sigma SBF -
+
(T ) \cap E0(T ) and \sigma SBF -
+
(S) \cap E0(S) are nonempty. This contradicts the fact that
T and S satisfies property (Sw) (see Lemma 3.1).
Theorem 3.1 is proved.
4. Perturbations. Let [T, S] = TS - ST denote the commutator of the operators T and S. If
Q1 \in \scrB (\scrX ) and Q2 \in \scrB (\scrY ) are quasinilpotent operators such that [Q1, T ] = [Q2, S] = 0 for some
operators T \in \scrB (\scrX ) and S \in \scrB (\scrY ), then
(T +Q1)\otimes (S +Q2) = (T \otimes S) +Q,
where Q = Q1 \otimes S + T \otimes Q2 +Q1 \otimes Q2 \in \scrB (\scrX \otimes \scrY ) is quasinilpotent operator.
Recall that T \in \scrB (\scrX ) is finitely isoloid if \lambda \in \mathrm{i}\mathrm{s}\mathrm{o}\sigma (T ) implies \lambda \in E0(T ).
Theorem 4.1. Let T \in \scrB (\scrX ) and S \in \scrB (\scrY ) having SVEP and let Q1 \in \scrB (\scrX ) and Q2 \in \scrB (\scrY )
be quasinilpotent operators such that [Q1, T ] = [Q2, S] = 0. If T \otimes S is finitely isoloid, then T \otimes S
satisfies property (Sw) implies (T +Q1)\otimes (S +Q2) satisfies property (Sw).
Proof. Recall that \sigma ((T +Q1)\otimes (S+Q2)) = \sigma (T \otimes S), \sigma a((T +Q1)\otimes (S+Q2)) = \sigma a(T \otimes S),
\sigma SBF -
+
((T + Q1) \otimes (S + Q2)) = \sigma SBF -
+
(T \otimes S) and that the perturbation of an operator by a
commuting quasinilpotent has SVEP if and only if the operator has SVEP. If T \otimes S satisfies property
(Sw), then
E0(T \otimes S) = \sigma (T \otimes S) \setminus \sigma SBF -
+
(T \otimes S) =
= \sigma ((T +Q1)\otimes (S +Q2)) \setminus \sigma SBF -
+
((T +Q1)\otimes (S +Q2)).
We prove E0(T \otimes S) = E0((T +Q1)\otimes (S +Q2)). Observe that if \lambda \in \mathrm{i}\mathrm{s}\mathrm{o}\sigma (T \otimes S), then T \ast \otimes S\ast
has SVEP at \lambda ; equivalently, (T \ast + Q\ast
1) \otimes (S\ast + Q\ast
2) has SVEP at \lambda . Let \lambda \in E0(T \otimes S). Then
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STABILITY OF VERSIONS OF THE WEYL-TYPE THEOREMS UNDER TENSOR PRODUCT 547
\lambda \in \sigma ((T +Q1)\otimes (S +Q2)) \setminus \sigma SBF -
+
((T +Q1)\otimes (S +Q2)). Since (T +Q1)
\ast \otimes (S +Q2)
\ast has
SVEP at \lambda , it follows that \lambda /\in \sigma w((T +Q1)\otimes (S +Q2)) and \lambda \in \mathrm{i}\mathrm{s}\mathrm{o} ((T +Q1)\otimes (S +Q2)). Thus
\lambda \in E0((T + Q1) \otimes (S + Q2)). Hence E0(T \otimes S) \subseteq E0((T + Q1) \otimes (S + Q2)). Conversely, if
\lambda \in E0((T +Q1)\otimes (S+Q2)), then \lambda \in \mathrm{i}\mathrm{s}\mathrm{o} (T \otimes S), and this, since T \otimes S is finitely isoloid, implies
that \lambda \in E0(T \otimes S). Hence E0((T +Q1)\otimes (S +Q2)) \subseteq E0(T \otimes S).
Theorem 4.1 is proved.
From [6], we recall that an operator R \in \scrB (\scrX ) is said to be Riesz if R - \lambda I is Fredholm for
every non-zero complex number \lambda .
For a bounded operator T on \scrX , we denote by E0f (T ) the set of isolated points \lambda of \sigma (T ) such
that \mathrm{k}\mathrm{e}\mathrm{r}(T - \lambda I) is finite-dimensional. Evidently, E0(T ) \subseteq E0f (T ).
Lemma 4.1. Let T be a bounded operator on \scrX . If R is a Riesz operator that commutes with
T, then
E0(T +R) \cap \sigma (T ) \subseteq \mathrm{i}\mathrm{s}\mathrm{o}\sigma (T ).
Proof. Clearly,
E0(T +R) \cap \sigma (T ) \subseteq E0f (T +R) \cap \sigma (T )
and by Lemma 2.3 of [16] the last set contained in \mathrm{i}\mathrm{s}\mathrm{o}\sigma (T ).
Lemma 4.1 is proved.
Now we consider the perturbations by commuting Riesz operators. Let T,R \in \scrB (\scrX ) be such
that R is Riesz and [T,R] = 0. The tensor product T \otimes R is not a Riesz operator (the Fredholm
spectrum \sigma F (T \otimes R) = \sigma (T )\sigma F (R) \cup \sigma F (T )\sigma (R) = \sigma F (T )\sigma (R) = \{ 0\} for a particular choice of
T only). However, \sigma w (also, \sigma b) is stable under perturbation by commuting Riesz operators [23],
and so T satisfies Browder’s theorem if and only if T + R satisfies Browder’s theorem. Thus,
if \sigma (T ) = \sigma (T + R) for a certain choice of operators T,R \in \scrB (\scrX ) (such that R is Riesz and
[T,R] = 0), then
\pi 0(T ) = \sigma (T ) \setminus \sigma w(T ) = \sigma (T +R) \setminus \sigma w(T +R) = \pi 0(T +R),
where \pi 0(T ) is the set of \lambda \in \mathrm{i}\mathrm{s}\mathrm{o}\sigma (T ) which are finite rank poles of the resolvent of T. If we now
suppose additionally that T satisfies property (Sw), then
E0(T ) = \sigma (T ) \setminus \sigma SBF -
+
(T ) = \sigma (T ) \setminus \sigma w(T ) = \sigma (T +R) \setminus \sigma w(T +R), (4.1)
and a necessary and sufficient condition for T+R to satisfy property (Sw) is that E0
a(T+R) = E0
a(T ).
One such condition, namely T is finitely isoloid.
Proposition 4.1. Let T,R \in \scrB (\scrX ), where R is Riesz, [T,R] = 0 and T is finitely isoloid. Then
T satisfies property (Sw) implies T +R satisfies property (Sw).
Proof. Observe that if T obeys property (Sw), then identity (4.1) holds. Let \lambda \in E0(T ). Then it
follows from Lemma 4.1 that \lambda \in E0(T )\cap \sigma (T ) = E0(T +R - R) \subseteq \mathrm{i}\mathrm{s}\mathrm{o}\sigma (T +R) and so T \ast +R\ast
has SVEP at \lambda . Since \lambda \in \sigma (T + R) \setminus \sigma w(T + R), T \ast + R\ast has SVEP at \lambda implies T + R - \lambda is
Fredholm of index 0 and so \lambda \in E0(T +R). Thus E0(T ) \subseteq E0(T +R). Now let \lambda \in E0(T +R).
Then \lambda \in E0(T + R) \cap \sigma (T + R) = E0(T + R) \cap \sigma (T ) \subseteq \mathrm{i}\mathrm{s}\mathrm{o}\sigma (T ), which by the finite isoloid
property of T implies \lambda \in E0(T ). Hence E0(T +R) \subseteq E0(T ).
Proposition 4.1 is proved.
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548 M. H. M. RASHID, T. PRASAD
Theorem 4.2. Let T \in \scrB (\scrX ) and S \in \scrB (\scrX ) be finitely isoloid operators which satisfy property
(Sw). If R1 \in \scrB (\scrX ) and R2 \in \scrB (\scrY ) are Riesz operators such that [T,R1] = [S,R2] = 0, \sigma (T +
+R1) = \sigma (T ) and \sigma (S+R2) = \sigma (S), then T\otimes S satisfies property (Sw) implies (T+R1)\otimes (S+R2)
satisfies property (Sw) if and only if Browder’s theorem transforms from T +R1 and S+R2 to their
tensor product.
Proof. The hypotheses imply (by Proposition 4.1) that both T +R1 and S +R2 satisfy property
(Sw). Suppose that T \otimes B satisfies property (Sw). Then \sigma (T \otimes B) \setminus \sigma SBF -
+
(T \otimes S) = E0(T \otimes S).
Evidently T \otimes B satisfies Browder’s theorem, and so the hypothesis T and B satisfy property (Sw)
implies that Browder’s theorem transfers from T and S to T \otimes S. Furthermore, since, \sigma (T +R1) =
= \sigma (T ), \sigma (S +R2) = \sigma (S), and \sigma w is stable under perturbations by commuting Riesz operators,
\sigma SBF -
+
(T \otimes S) = \sigma w(T \otimes S) = \sigma (T )\sigma w(S) \cup \sigma w(T )\sigma (S) =
= \sigma (T +R1)\sigma w(S +R2) \cup \sigma w(T +R1)\sigma (S +R2) =
= \sigma (T +R1)\sigma SBF -
+
(S +R2) \cup \sigma SBF -
+
(T +R1)\sigma (S +R2).
Suppose now that Browder’s theorem transfers from T + R1 and S + R2 to (T + R1)\otimes (S + R2).
Then
\sigma w(T \otimes S) = \sigma w((T +R1)\otimes (S +R2))
and
E0(T \otimes S) = \sigma ((T +R1)\otimes (S +R2)) \setminus \sigma w((T +R1)\otimes (S +R2)).
Let \lambda \in E0(T \otimes S). Then \lambda \not = 0, and hence there exist \mu \in \sigma (T + R1) \setminus \sigma w(T + R1) and
\nu \in \sigma (S+R2) \setminus \sigma w(S+R2) such that \lambda = \mu \nu . As observed above, both T +R1 and S+R2 satisfy
property (Sw); hence \mu \in E0(S + R1) and \nu \in E0(S + R2). This, since \lambda \in \sigma (T \otimes S) = \sigma ((T +
+R1)\otimes (S+R2)), implies \lambda \in E0((T+R1)\otimes (S+R2)). Conversely, if \lambda \in E0((T+R1)\otimes (S+R2)),
then \lambda \not = 0 and there exist \mu \in E0(T + R1) \subseteq \mathrm{i}\mathrm{s}\mathrm{o}\sigma (T ) and \nu \in E0(S + R2) \subseteq \mathrm{i}\mathrm{s}\mathrm{o}\sigma (S) such
that \lambda = \mu \nu . Recall that E0((T + R1) \otimes (S + R2)) \subseteq E0(T + R1)E
0(S + R2). Since T and S
are finite isoloid, \mu \in E0(T ) and \nu \in E0(S). Hence, since \sigma ((T + R1) \otimes (S + R2)) = \sigma (T \otimes S),
\lambda = \mu \nu \in E0(T \otimes S). To complete the proof, we observe that if the implication of the statement of
the theorem holds, then (necessarily) (T +R1)\otimes (S +R2) satisfies Browder’s theorem. This, since
T + R1 and S + R2 satisfy Browder’s theorem, implies Browder’s theorem transfers from T + R1
and S +R2 to (T +R1)\otimes (S +R2).
Theorem 4.2 is proved.
5. Property (\bfitS \bfitw ) for direct sum. Let \scrH and \scrK be infinite-dimensional Hilbert spaces. In this
section we show that if T and S are two operators on \scrH and \scrK respectively and at least one of them
satisfies property (Sw) then their direct sum T \oplus S obeys property (Sw). We also explore various
conditions on T and S to ensure that T \oplus S satisfies property (Sw).
Theorem 5.1. Suppose that property (Sw) holds for T \in \scrB (\scrH ) and S \in \scrB (\scrK ). If T and S are
isoloid and \sigma SBF -
+
(T \oplus S) = \sigma SBF -
+
(T ) \cup \sigma SBF -
+
(S), then property (Sw) holds for T \oplus S.
Proof. We know that \sigma (T \oplus S) = \sigma (T )\cup \sigma (S) for any pairs of operators. If T and S are isoloid,
then
E0(T \oplus S) =
\bigl[
E0(T ) \cap \rho (S)
\bigr]
\cup
\bigl[
\rho (T ) \cap E0(S)
\bigr]
\cup
\bigl[
E0(T ) \cap E0(S)
\bigr]
,
where \rho (.) = \BbbC \setminus \sigma (.).
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STABILITY OF VERSIONS OF THE WEYL-TYPE THEOREMS UNDER TENSOR PRODUCT 549
If property (Sw) holds for T and S, then
[\sigma (T ) \cup \sigma (S)] \setminus
\Bigl[
\sigma SBF -
+
(T ) \cup \sigma SBF -
+
(S)
\Bigr]
=
=
\bigl[
E0(T ) \cap \rho (S)
\bigr]
\cup
\bigl[
\rho (T ) \cap E0(S)
\bigr]
\cup
\bigl[
E0(T ) \cap E0(S)
\bigr]
.
Thus, E0(T \oplus S) = [\sigma (T ) \cup \sigma (S)] \setminus
\Bigl[
\sigma SBF -
+
(T ) \cup \sigma SBF -
+
(S)
\Bigr]
.
If \sigma SBF -
+
(T \oplus S) = \sigma SBF -
+
(T ) \cup \sigma SBF -
+
(S), then
E0(T \oplus S) = \sigma (T \oplus S) \setminus \sigma SBF -
+
(T \oplus S).
Hence property (Sw) holds for T \oplus S.
Theorem 5.1 is proved.
Theorem 5.2. Suppose that T \in \scrB (\scrH ) such that \mathrm{i}\mathrm{s}\mathrm{o}\sigma (T ) = \varnothing and S \in \scrB (\scrK ) satisfies property
(Sw). If \sigma SBF -
+
(T \oplus S) = \sigma (T ) \cup \sigma SBF -
+
(S), then property (Sw) holds for T \oplus S.
Proof. We know that \sigma (T \oplus S) = \sigma (T ) \cup \sigma (S) for any pairs of operators. Then
\sigma (T \oplus S) \setminus \sigma SBF -
+
(T \oplus S) = [\sigma (T ) \cup \sigma (S)] \setminus
\Bigl[
\sigma (T ) \cup \sigma SBF -
+
(S)
\Bigr]
=
= \sigma (S) \setminus
\Bigl[
\sigma (T ) \cup \sigma SBF -
+
(S)
\Bigr]
=
=
\Bigl[
\sigma (S) \setminus \sigma SBF -
+
(S)
\Bigr]
\setminus \sigma (T ) = E0(S) \cap \rho (T ).
If \mathrm{i}\mathrm{s}\mathrm{o}\sigma (T ) = \varnothing it implies that \sigma (T ) = \mathrm{a}\mathrm{c}\mathrm{c}\sigma (T ), where \mathrm{a}\mathrm{c}\mathrm{c}\sigma (T ) = \sigma (T ) \setminus \mathrm{i}\mathrm{s}\mathrm{o}\sigma (T ) is the set of all
accumulation points of \sigma (T ). Thus we have
\mathrm{i}\mathrm{s}\mathrm{o}\sigma (T \oplus S) = [\mathrm{i}\mathrm{s}\mathrm{o}\sigma (T ) \cup \mathrm{i}\mathrm{s}\mathrm{o}\sigma (S)] \setminus [(\mathrm{i}\mathrm{s}\mathrm{o}\sigma (T ) \cap \mathrm{a}\mathrm{c}\mathrm{c}\sigma (S)) \cup (\mathrm{a}\mathrm{c}\mathrm{c}\sigma (T ) \cap \mathrm{i}\mathrm{s}\mathrm{o}\sigma (S))] =
= [\mathrm{i}\mathrm{s}\mathrm{o}\sigma (T ) \setminus \mathrm{a}\mathrm{c}\mathrm{c}\sigma (S)] \cup [\mathrm{i}\mathrm{s}\mathrm{o}\sigma (S) \setminus \mathrm{a}\mathrm{c}\mathrm{c}\sigma (T )] = \mathrm{i}\mathrm{s}\mathrm{o}\sigma (S) \setminus \sigma (T ) = \mathrm{i}\mathrm{s}\mathrm{o}\sigma (S) \cap \rho (T ).
We know that \sigma p(T \oplus S) = \sigma p(T )\cup \sigma p(S) and \alpha (T \oplus S) = \alpha (T ) +\alpha (S) for any pairs of operators
T and S, so that
\sigma PF (T \oplus S) = \{ \lambda \in \sigma PF (T ) \cup \sigma PF (S)\alpha (T - \lambda I) + \alpha (S - \lambda I) < \infty \} .
Therefore,
E0(T \oplus S) = \mathrm{i}\mathrm{s}\mathrm{o}\sigma (T \oplus S) \cap \sigma PF (T \oplus S) = \mathrm{i}\mathrm{s}\mathrm{o}\sigma (S) \cap \rho (T ) \cap \sigma PF (S) = E0(S) \cap \rho (T ).
Thus \sigma (T \oplus S) \setminus \sigma SBF -
+
(T \oplus S) = E0(T \oplus S). Hence T \oplus S satisfies property (Sw).
Theorem 5.2 is proved.
Corollary 5.1. Suppose that T \in \scrB (\scrH ) is such that \mathrm{i}\mathrm{s}\mathrm{o}\sigma (T ) = \varnothing and S \in \scrB (\scrH ) satisfies
property (Sw) with \mathrm{i}\mathrm{s}\mathrm{o}\sigma (S)\cap \sigma p(S) = \varnothing , and \Delta g
a(T \oplus S) = \varnothing , then T \oplus S satisfies property (Sw).
Proof. Since S satisfies property (Sw), therefore given condition \mathrm{i}\mathrm{s}\mathrm{o}\sigma (S) \cap \sigma p(S) = \varnothing implies
that \sigma (S) = \sigma SBF -
+
(S). Now \Delta g
a(T \oplus S) = \varnothing gives that \sigma SBF -
+
(T \oplus S) = \sigma (T \oplus S) = \sigma (T ) \cup
\cup \sigma SBF -
+
(S). Thus from Theorem 5.2, we have that T \oplus S satisfies property (Sw).
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 4
550 M. H. M. RASHID, T. PRASAD
Corollary 5.2. Suppose that T \in \scrB (\scrH ) is such that \mathrm{i}\mathrm{s}\mathrm{o}\sigma (T ) \cup \Delta g
a(T ) = \varnothing and S \in \scrB (\scrK )
satisfies property (Sw). If \sigma SBF -
+
(T \oplus S) = \sigma SBF -
+
(T ) \cup \sigma SBF -
+
(S), then T \oplus S satisfies proper-
ty (Sw).
Theorem 5.3. Let T \in \scrB (\scrH ) be an isoloid operator that satisfies property (Sw). If S \in \scrB (\scrK )
is a normal operator satisfies property (Sw), then property (Sw) holds for T \oplus S.
Proof. If S is normal, then both S and S\ast have SVEP, and \mathrm{i}\mathrm{n}\mathrm{d} (S - \lambda I) = 0 for every \lambda such
that S - \lambda I is a B-Fredholm. Observe that \lambda /\in \sigma SBF -
+
(T \oplus S) if and only if S - \lambda I \in SBF+(K)
and T - \lambda I \in SBF+(H) and \mathrm{i}\mathrm{n}\mathrm{d} (T - \lambda I) + \mathrm{i}\mathrm{n}\mathrm{d} (S - \lambda I) = \mathrm{i}\mathrm{n}\mathrm{d} (T - \lambda I) \leq 0 if and only if
\lambda /\in \Delta g
a(T ) \cap \Delta g
a(S). Hence \sigma SBF -
+
(T \oplus S) = \sigma SBF -
+
(T ) \cup \sigma SBF -
+
(S). It is well known that
the isolated points of the approximate point spectrum of a normal operator are simple poles of the
resolvent of the operator implies that S is isoloid. So the result follows now from Theorem 5.1.
Theorem 5.3 is proved.
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Received 16.03.13,
after revision — 18.11.15
ISSN 1027-3190. Укр. мат. журн., 2016, т. 68, № 4
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| id | umjimathkievua-article-1859 |
| institution | Ukrains’kyi Matematychnyi Zhurnal |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2026-03-24T02:14:05Z |
| publishDate | 2016 |
| publisher | Institute of Mathematics, NAS of Ukraine |
| record_format | ojs |
| resource_txt_mv | umjimathkievua/e7/958223b665e0c8c8e3a2cda760b290e7.pdf |
| spelling | umjimathkievua-article-18592019-12-05T09:29:54Z Stability of versions of the Weyl-type theorems under tensor product Стабiльнiсть рiзних версiй теорем типу Вейля для тензорного добутку Prasad, T. Rashid, M. H. M. Прасад, Т. Рашид, М. Х. М. We study the transformation versions of the Weyl-type theorems from operators $T$ and $S$ for their tensor product $T \otimes S$ in the infinite-dimensional space setting. Вивчаються трансформованi версiї теорем типу Вейля для операторiв $T$ i $S$ та їх тензорного добутку $T \otimes S$ у нескiнченновимiрнiй постановцi. Institute of Mathematics, NAS of Ukraine 2016-04-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/1859 Ukrains’kyi Matematychnyi Zhurnal; Vol. 68 No. 4 (2016); 542-550 Український математичний журнал; Том 68 № 4 (2016); 542-550 1027-3190 en https://umj.imath.kiev.ua/index.php/umj/article/view/1859/841 Copyright (c) 2016 Prasad T.; Rashid M. H. M. |
| spellingShingle | Prasad, T. Rashid, M. H. M. Прасад, Т. Рашид, М. Х. М. Stability of versions of the Weyl-type theorems under tensor product |
| title | Stability of versions of the Weyl-type theorems under tensor product |
| title_alt | Стабiльнiсть рiзних версiй теорем типу Вейля для тензорного добутку |
| title_full | Stability of versions of the Weyl-type theorems under tensor product |
| title_fullStr | Stability of versions of the Weyl-type theorems under tensor product |
| title_full_unstemmed | Stability of versions of the Weyl-type theorems under tensor product |
| title_short | Stability of versions of the Weyl-type theorems under tensor product |
| title_sort | stability of versions of the weyl-type theorems under tensor product |
| url | https://umj.imath.kiev.ua/index.php/umj/article/view/1859 |
| work_keys_str_mv | AT prasadt stabilityofversionsoftheweyltypetheoremsundertensorproduct AT rashidmhm stabilityofversionsoftheweyltypetheoremsundertensorproduct AT prasadt stabilityofversionsoftheweyltypetheoremsundertensorproduct AT rašidmhm stabilityofversionsoftheweyltypetheoremsundertensorproduct AT prasadt stabilʹnistʹriznihversijteoremtipuvejlâdlâtenzornogodobutku AT rashidmhm stabilʹnistʹriznihversijteoremtipuvejlâdlâtenzornogodobutku AT prasadt stabilʹnistʹriznihversijteoremtipuvejlâdlâtenzornogodobutku AT rašidmhm stabilʹnistʹriznihversijteoremtipuvejlâdlâtenzornogodobutku |