scholarly journals Spectrum localization of a perturbed operator in a strip and applications

2021 ◽  
Vol 41 (3) ◽  
pp. 395-412
Author(s):  
Michael Gil'
Keyword(s):  
Hilbert Space ◽  
Integral Operators ◽  
Bounded Operators ◽  
Schmidt Operator ◽  
Spectrum Localization

Let \(A\) and \(\tilde{A}\) be bounded operators in a Hilbert space. We consider the following problem: let the spectrum of \(A\) lie in some strip. In what strip the spectrum of \(\tilde{A}\) lies if \(A\) and \(\tilde{A}\) are "close"? Applications of the obtained results to integral operators and matrices are also discussed. In addition, we apply our perturbation results to approximate the spectral strip of a Hilbert-Schmidt operator by the spectral strips of finite matrices.

scholarly journals Correctness conditions for high-order differential equations with unbounded coefficients

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Kordan N. Ospanov
Keyword(s):  
Differential Equation ◽  
Hilbert Space ◽  
Existence And Uniqueness ◽  
Sufficient Conditions ◽  
Integral Operators ◽  
Order Linear Differential Equation ◽  
Unbounded Coefficients ◽  
Weighted Norms ◽  
Uniqueness Of The Solution ◽  
Linear Differential

AbstractWe give some sufficient conditions for the existence and uniqueness of the solution of a higher-order linear differential equation with unbounded coefficients in the Hilbert space. We obtain some estimates for the weighted norms of the solution and its derivatives. Using these estimates, we show the conditions for the compactness of some integral operators associated with the resolvent.

scholarly journals TENSOR EXTENSION PROPERTIES OF C(K)-REPRESENTATIONS AND APPLICATIONS TO UNCONDITIONALITY

2010 ◽  
Vol 88 (2) ◽  
pp. 205-230 ◽  
Author(s):  
CHRISTOPH KRIEGLER ◽  
CHRISTIAN LE MERDY
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Unconditional Basis ◽  
Compact Set ◽  
Bounded Operators ◽  
Space Setting ◽  
Uniformly Bounded

AbstractLet K be any compact set. The C*-algebra C(K) is nuclear and any bounded homomorphism from C(K) into B(H), the algebra of all bounded operators on some Hilbert space H, is automatically completely bounded. We prove extensions of these results to the Banach space setting, using the key concept ofR-boundedness. Then we apply these results to operators with a uniformly bounded H∞-calculus, as well as to unconditionality on Lp. We show that any unconditional basis on Lp ‘is’ an unconditional basis on L2 after an appropriate change of density.

The Spectra of Semi-Normal Singular Integral Operators

1970 ◽  
Vol 22 (1) ◽  
pp. 134-150 ◽  
Author(s):  
C. R. Putnam
Keyword(s):  
Hilbert Space ◽  
Singular Integral ◽  
Integral Operators ◽  
Singular Integral Operators ◽  
Cauchy Principal ◽  
Cauchy Principal Value ◽  
Principal Value ◽  
Image Position ◽  
Adjoint Operators

Suppose that(1.1)and define the bounded self-adjoint operators H and J on the Hilbert space L2(0, 1) by(1.2)the integral being a Cauchy principal valueIt is seen that(1.3)or, equivalently,(1.4)Since (Cƒ, ƒ) = π–1|(ƒ, ϕ)|2 ≧ 0, A is semi-normal. (For a discussion of such operators, see [4].)

Three Test Problems for Quasisimilarity

1987 ◽  
Vol 39 (4) ◽  
pp. 880-892 ◽  
Author(s):  
Hari Bercovici
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Linear Operators ◽  
Structure Theory ◽  
Test Problems ◽  
Unitary Equivalence ◽  
Bounded Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Densely Defined

Kaplansky proposed in [7] three problems with which to test the adequacy of a proposed structure theory of infinite abelian groups. These problems can be rephrased as test problems for a structure theory of operators on Hilbert space. Thus, R. Kadison and I. Singer answered in [6] these test problems for the unitary equivalence of operators. We propose here a study of these problems for quasisimilarity of operators on Hilbert space. We recall first that two (bounded, linear) operators T and T′ acting on the Hilbert spaces and , are said to be quasisimilar if there exist bounded operators and with densely defined inverses, satisfying the relations T′X = XT and TY = YT′. The fact that T and T′ are quasisimilar is indicated by T ∼ T′. The problems mentioned above can now be formulated as follows.

On the Definition of C*-Algebras II

1985 ◽  
Vol 37 (4) ◽  
pp. 664-681 ◽  
Author(s):  
Zoltán Magyar ◽  
Zoltán Sebestyén
Keyword(s):  
Hilbert Space ◽  
Banach Algebra ◽  
Representation Theorem ◽  
Banach Algebras ◽  
Bounded Operators ◽  
Fundamental Representation ◽  
C Algebras ◽  
Definition Of ◽  
Image Position

The theory of noncommutative involutive Banach algebras (briefly Banach *-algebras) owes its origin to Gelfand and Naimark, who proved in 1943 the fundamental representation theorem that a Banach *-algebra with C*-condition(C*)is *-isomorphic and isometric to a norm-closed self-adjoint subalgebra of all bounded operators on a suitable Hilbert space.At the same time they conjectured that the C*-condition can be replaced by the B*-condition.(B*)In other words any B*-algebra is actually a C*-algebra. This was shown by Glimm and Kadison [5] in 1960.

Asymptotic normality of sums of Hilbert space valued random elements

2019 ◽  
Vol 0 (0) ◽  
Author(s):  
Alfredas Račkauskas
Keyword(s):  
Hilbert Space ◽  
Asymptotic Normality ◽  
Separable Hilbert Space ◽  
Linear Process ◽  
Weighted Sums ◽  
Bounded Operators ◽  
Linear Processes ◽  
Summation Methods ◽  
Random Elements

Abstract We investigate the asymptotic normality of distributions of the sequence {\sum_{k\in\mathbb{Z}}u_{n,k}X_{k}} , {n\in\mathbb{N}} , where {(X_{k},k\in\mathbb{Z})} either is a sequence of i.i.d. random elements or constitutes a linear process with i.i.d. innovations in a separable Hilbert space. The weights {(u_{n,k})} are in general a family of linear bounded operators. This model includes operator weighted sums of Hilbert space valued linear processes, operator-wise discounted sums in a Hilbert space as well some extensions of classical summation methods.

Representation of Banach Algebras with an Involution

1957 ◽  
Vol 9 ◽  
pp. 435-442
Author(s):  
J. A. Schatz
Keyword(s):  
Hilbert Space ◽  
Banach Spaces ◽  
Banach Algebras ◽  
Bounded Operators ◽  
Uniformly Closed

In 1943 Gelfand and Neumark (3) characterized uniformly closed self-adjoint algebras of bounded operators on a Hilbert space as Banach algebras with an involution (a conjugate linear anti-isomorphism of period two) satisfying several additional conditions. The main purpose of this paper is to point out that if we consider algebras of bounded operators on complex Banach spaces more general than Hilbert space, then we can represent a larger class of algebras by essentially the same methods.

FROM QUANTUM FIELDS TO LOCAL VON NEUMANN ALGEBRAS

1992 ◽  
Vol 04 (spec01) ◽  
pp. 15-47 ◽  
Author(s):  
H.J. BORCHERS ◽  
JAKOB YNGVASON
Keyword(s):  
Hilbert Space ◽  
Von Neumann Algebras ◽  
Sufficient Conditions ◽  
Bounded Operators ◽  
Unbounded Operators ◽  
Von Neumann ◽  
Energy Bounds ◽  
Local Algebras ◽  
Quantized Fields ◽  
Necessary And Sufficient

The subject of the paper is an old problem of the general theory of quantized fields: When can the unbounded operators of a Wightman field theory be associated with local algebras of bounded operators in the sense of Haag? The paper reviews and extends previous work on this question, stressing its connections with a noncommutive generalization of the classical Hamburger moment problem. Necessary and sufficient conditions for the existence of a local net of von Neumann algebras corresponding to a given Wightman field are formulated in terms of strengthened versions of the usual positivity property of Wightman functionals. The possibility that the local net has to be defined in an enlarged Hilbert space cannot be ruled out in general. Under additional hypotheses, e.g., if the field operators obey certain energy bounds, such an extension of the Hilbert space is not necessary, however. In these cases a fairly simple condition for the existence of a local net can be given involving the concept of “central positivity” introduced by Powers. The analysis presented here applies to translationally covariant fields with an arbitrary number of components, whereas Lorentz covariance is not needed. The paper contains also a brief discussion of an approach to noncommutative moment problems due to Dubois-Violette, and concludes with some remarks on modular theory for algebras of unbounded operators.

scholarly journals A SIMPLE PROOF OF THE FUNDAMENTAL THEOREM ABOUT ARVESON SYSTEMS

2006 ◽  
Vol 09 (02) ◽  
pp. 305-314 ◽  
Author(s):  
MICHAEL SKEIDE
Keyword(s):  
Hilbert Space ◽  
Simple Proof ◽  
Fundamental Theorem ◽  
Hilbert Module ◽  
Bounded Operators ◽  
Infinite Dimensional ◽  
Infinite Dimensional Hilbert Space ◽  
Product Systems ◽  
Dimensional Hilbert Space ◽  
The One

With every E0-semigroup (acting on the algebra of of bounded operators on a separable infinite-dimensional Hilbert space) there is an associated Arveson system. One of the most important results about Arveson systems is that every Arveson system is the one associated with an E0-semigroup. In these notes we give a new proof of this result that is considerably simpler than the existing ones and allows for a generalization to product systems of Hilbert module (to be published elsewhere).

scholarly journals Schatten's theorems on functionally defined Schur algebras

2005 ◽  
Vol 2005 (14) ◽  
pp. 2175-2193 ◽  
Author(s):  
Pachara Chaisuriya ◽  
Sing-Cheong Ong
Keyword(s):  
Hilbert Space ◽  
Banach Algebra ◽  
Bounded Operator ◽  
Trace Class ◽  
Compact Operators ◽  
Bounded Operators ◽  
Schur Algebras ◽  
Schur Product ◽  
Product Operation ◽  
Trace Class Operators

For each triple of positive numbersp,q,r≥1and each commutativeC*-algebraℬwith identity1and the sets(ℬ)of states onℬ, the set𝒮r(ℬ)of all matricesA=[ajk]overℬsuch thatϕ[A[r]]:=[ϕ(|ajk|r)]defines a bounded operator fromℓptoℓqfor allϕ∈s(ℬ)is shown to be a Banach algebra under the Schur product operation, and the norm‖A‖=‖|A|‖p,q,r=sup{‖ϕ[A[r]]‖1/r:ϕ∈s(ℬ)}. Schatten's theorems about the dual of the compact operators, the trace-class operators, and the decomposition of the dual of the algebra of all bounded operators on a Hilbert space are extended to the𝒮r(ℬ)setting.

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