Weyl’s theorem for direct sums

2007 ◽  
Vol 44 (2) ◽  
pp. 275-290
Author(s):  
Bhagwati Duggal ◽  
Carlos Kubrusly
Keyword(s):  
Hilbert Space ◽  
Sufficient Conditions ◽  
The Other ◽  
Weyl’S Theorem ◽  
Direct Sums ◽  
Direct Sum ◽  
Hilbert Space Operators ◽  
Isolated Points

Let T and S be Hilbert space operators such that Weyl’s theorem holds for both of them. In general, it does not follow that Weyl’s theorem holds for the direct sum T ⊕ S . We give asymmetric sufficient conditions on T and S to ensure that the direct sum T ⊕ S satisfies Weyl’s theorem. It is assumed that just one of the direct summands satisfies Weyl’s theorem but is not necessarily isoloid, while the other has no isolated points in its spectrum.

scholarly journals Structure of n-quasi left m-invertible and related classes of operators

2020 ◽  
Vol 53 (1) ◽  
pp. 249-268
Author(s):  
Bhagwati Prashad Duggal ◽  
In Hyun Kim
Keyword(s):  
Hilbert Space ◽  
Positive Integer ◽  
Direct Sum ◽  
Hilbert Space Operators ◽  
Isolated Points ◽  
Elementary Operators ◽  
Adjoint Operators ◽  
Number Of Classes ◽  
Symmetric Operators ◽  
Power Bounded

AbstractGiven Hilbert space operators T,S\in B( {\mathcal H} ), let \text{Δ} and \delta \in B(B( {\mathcal H} )) denote the elementary operators {\text{Δ}}_{T,S}(X)=({L}_{T}{R}_{S}-I)(X)=TXS-X and {\delta }_{T,S}(X)=({L}_{T}-{R}_{S})(X)=TX-XS. Let d=\text{Δ} or \delta . Assuming T commutes with {S}^{\ast }, and choosing X to be the positive operator {S}^{\ast n}{S}^{n} for some positive integer n, this paper exploits properties of elementary operators to study the structure of n-quasi {[}m,d]-operators {d}_{T,S}^{m}(X)=0 to bring together, and improve upon, extant results for a number of classes of operators, such as n-quasi left m-invertible operators, n-quasi m-isometric operators, n-quasi m-self-adjoint operators and n-quasi (m,C) symmetric operators (for some conjugation C of {\mathcal H} ). It is proved that {S}^{n} is the perturbation by a nilpotent of the direct sum of an operator {S}_{1}^{n}={\left(S{|}_{\overline{{S}^{n}( {\mathcal H} )}}\right)}^{n} satisfying {d}_{{T}_{1},{S}_{1}}^{m}({I}_{1})=0, {{T}_{1}=T}_{\overline{{S}^{n}( {\mathcal H} )}}, with the 0 operator; if S is also left invertible, then {S}^{n} is similar to an operator B such that {d}_{{B}^{\ast },B}^{m}(I)=0. For power bounded S and T such that S{T}^{\ast }-{T}^{\ast }S=0 and {\text{Δ}}_{T,S}({S}^{\ast n}{S}^{n})=0, S is polaroid (i.e., isolated points of the spectrum are poles). The product property, and the perturbation by a commuting nilpotent property, of operators T,S satisfying {d}_{T,S}^{m}(I)=0, given certain commutativity properties, transfers to operators satisfying {S}^{\ast n}{d}_{T,S}^{m}(I){S}^{n}=0.

scholarly journals Algebraic entropies, Hopficity and co-Hopficity of direct sums of Abelian Groups

2015 ◽  
Vol 3 (1) ◽  
Author(s):  
Brendan Goldsmith ◽  
Ketao Gong
Keyword(s):  
Abelian Groups ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Direct Sums ◽  
Direct Sum ◽  
Zero Entropy ◽  
Necessary And Sufficient

AbstractNecessary and sufficient conditions to ensure that the direct sum of two Abelian groups with zero entropy is again of zero entropy are still unknown; interestingly the same problem is also unresolved for direct sums of Hopfian and co-Hopfian groups.We obtain sufficient conditions in some situations by placing restrictions on the homomorphisms between the groups. There are clear similarities between the various cases but there is not a simple duality involved.

scholarly journals Two conditions for subnormality of unbounded operators

2001 ◽  
Vol 43 (1) ◽  
pp. 23-28
Author(s):  
Jan Niechwiej
Keyword(s):  
Hilbert Space ◽  
Sufficient Conditions ◽  
Bounded Operators ◽  
Unbounded Operators ◽  
The Real ◽  
Completely Monotonic ◽  
Binomial Expansion ◽  
Hilbert Space Operators ◽  
The One

We give two new sufficient conditions for unbounded Hilbert space operators to be subnormal. The first assumes that the sequence //Tnf//2 on a suitable subset of the domain is completely monotonic, the second is similar to the one given by Lambert in [3] for bounded operators and involves the sequence of binomial expansion of the real part of the operator.

scholarly journals On the closure of the sum of closed subspaces

2001 ◽  
Vol 26 (5) ◽  
pp. 257-267 ◽  
Author(s):  
Irwin E. Schochetman ◽  
Robert L. Smith ◽  
Sze-Kai Tsui
Keyword(s):  
Hilbert Space ◽  
Quadratic Programming ◽  
Programming Problem ◽  
Sufficient Conditions ◽  
Optimal Solution ◽  
The Other ◽  
Quadratic Programming Problem ◽  
Orthogonal Projections ◽  
The Cost ◽  
Necessary And Sufficient

We give necessary and sufficient conditions for the sum of closed subspaces of a Hilbert space to be closed. Specifically, we show that the sum will be closed if and only if the angle between the subspaces is not zero, or if and only if the projection of either space into the orthogonal complement of the other is closed. We also give sufficient conditions for the sum to be closed in terms of the relevant orthogonal projections. As a consequence, we obtain sufficient conditions for the existence of an optimal solution to an abstract quadratic programming problem in terms of the kernels of the cost and constraint operators.

A Remark on the Krull-Schmidt-Azumaya Theorem

1970 ◽  
Vol 13 (4) ◽  
pp. 501-505 ◽  
Author(s):  
B. L. Osofsky
Keyword(s):  
Dedekind Domain ◽  
The Other ◽  
Group Algebras ◽  
Endomorphism Rings ◽  
Algebraic Integers ◽  
Direct Sums ◽  
Indecomposable Modules ◽  
Direct Sum ◽  
Other Hand

It is well known that if a module M is expressible as a direct sum of modules with local endomorphism rings, then such a decomposition is essentially unique. That is, if M = ⊕i∊IMi = ⊕j∊JNj then there is a bijection f: I → J such that Mi is isomorphic to Nf(i) for all i∊I (see [1]). On the other hand, a nonprincipal ideal in a Dedekind domain provides an example where such a theorem fails in the absence of the local hypothesis. Group algebras of certain groups over rings R of algebraic integers is another such example, where even the rank as R-modules of indecomposable summands of a module is not uniquely determined (see [2]). Both of these examples yield modules which are expressible as direct sums of two indecomposable modules in distinct ways. In this note we construct a family of rings which show that the number of summands in a representation of a module M as a direct sum of indecomposable modules is also not unique unless one has additional hypotheses.

scholarly journals Symmetrisable operators :Part III Hilbert space operators Symmetrisable by bounded operators

1964 ◽  
Vol 4 (1) ◽  
pp. 31-48 ◽  
Author(s):  
J. P. O. Silberstein
Keyword(s):  
Hilbert Space ◽  
The Other ◽  
Restricted Class ◽  
Bounded Operators ◽  
Hilbert Space Operators ◽  
Image Position ◽  
Symmetric Operators ◽  
Considerable Complexity

The fact that the most general symmetrisable operators in Hilbert Space do not possess a number of the desirable properties of such operators in unitary spaces makes it necessary to look for a more restricted class of operators. There are two reasons for our particular choice. In the first place many of the conditions introduced in the course of Part II concerned reltionships between the domain of the symmetrising operatorHand the domain and range of the symmetrisable operatorA. These conditions are now all automatically satisfied. The other reason is that the construction used in section 4 to relate symmetrisable operators to certain symmetric operators clearly required that eitherHorH−1was bounded. The case ofH−1bounded has already been dealt with in section 9 and shown to be fairly simple. The case in which H is bounded is clearly of considerable complexity, since we have already seen (example in proof of Theorem 10.6.) that the con |H| the bound ofHby

scholarly journals Operator equalities and Characterizations of Orthogonality in Pre-Hilbert C*-Modules

2021 ◽  
pp. 1-21
Author(s):  
Rasoul Eskandari ◽  
M. S. Moslehian ◽  
Dan Popovici
Keyword(s):  
Hilbert Space ◽  
Triangle Inequality ◽  
Sufficient Conditions ◽  
Negative Real Part ◽  
Necessary And Sufficient Conditions ◽  
Inner Product ◽  
Equality Case ◽  
Hilbert Space Operators ◽  
Necessary And Sufficient ◽  
The Relationship

Abstract In the first part of the paper, we use states on $C^{*}$ -algebras in order to establish some equivalent statements to equality in the triangle inequality, as well as to the parallelogram identity for elements of a pre-Hilbert $C^{*}$ -module. We also characterize the equality case in the triangle inequality for adjointable operators on a Hilbert $C^{*}$ -module. Then we give certain necessary and sufficient conditions to the Pythagoras identity for two vectors in a pre-Hilbert $C^{*}$ -module under the assumption that their inner product has a negative real part. We introduce the concept of Pythagoras orthogonality and discuss its properties. We describe this notion for Hilbert space operators in terms of the parallelogram law and some limit conditions. We present several examples in order to illustrate the relationship between the Birkhoff–James, Roberts, and Pythagoras orthogonalities, and the usual orthogonality in the framework of Hilbert $C^{*}$ -modules.

Lifting modules with finite internal exchange property and direct sums of hollow modules

2020 ◽  
pp. 2150189
Author(s):  
Yosuke Kuratomi
Keyword(s):  
Direct Summand ◽  
Sufficient Conditions ◽  
Exchange Property ◽  
Direct Sums ◽  
Perfect Ring ◽  
Direct Sum ◽  
Decomposition Formula ◽  
Direct Sum Decomposition ◽  
Extending Module ◽  
Necessary And Sufficient

A module [Formula: see text] is said to be lifting if, for any submodule [Formula: see text] of [Formula: see text], there exists a decomposition [Formula: see text] such that [Formula: see text] and [Formula: see text] is a small submodule of [Formula: see text]. A lifting module is defined as a dual concept of the extending module. A module [Formula: see text] is said to have the finite internal exchange property if, for any direct summand [Formula: see text] of [Formula: see text] and any finite direct sum decomposition [Formula: see text], there exists a direct summand [Formula: see text] of [Formula: see text] [Formula: see text] such that [Formula: see text]. This paper is concerned with the following two fundamental unsolved problems of lifting modules: “Classify those rings all of whose lifting modules have the finite internal exchange property” and “When is a direct sum of indecomposable lifting modules lifting?”. In this paper, we prove that any [Formula: see text]-square-free lifting module over a right perfect ring satisfies the finite internal exchange property. In addition, we give some necessary and sufficient conditions for a direct sum of hollow modules over a right perfect ring to be lifting with the finite internal exchange property.

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