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

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.

Eigenvalues of Finite Band-Width Hilbert Space Operators and Their Application to Orthogonal Polynomials

1989 ◽  
Vol 41 (1) ◽  
pp. 106-122 ◽  
Author(s):  
Attila Máté ◽  
Paul Nevai
Keyword(s):  
Hilbert Space ◽  
Orthogonal Polynomials ◽  
Real Number ◽  
Main Diagonal ◽  
Band Width ◽  
Positive Part ◽  
Hilbert Space Operators ◽  
Image Position ◽  
Finite Band

The main result of this paper concerns the eigenvalues of an operator in the Hilbert space l2that is represented by a matrix having zeros everywhere except in a neighborhood of the main diagonal. Write (c)+ for the positive part of a real number c, i.e., put (c+ = cif c≧ 0 and (c)+=0 otherwise. Then this result can be formulated as follows. Theorem 1.1. Let k ≧ 1 be an integer, and consider the operator S on l2 such that

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.

Strong Extensions vs. Weak Extensions of C*-Algebras

1978 ◽  
Vol 21 (2) ◽  
pp. 143-147
Author(s):  
S. J. Cho
Keyword(s):  
Hilbert Space ◽  
Partial Isometry ◽  
Compact Operators ◽  
Bounded Operators ◽  
Infinite Dimensional ◽  
C Algebras ◽  
Infinite Dimensional Hilbert Space ◽  
Dimensional Hilbert Space ◽  
Image Position ◽  
The Ideal

Let be a separable complex infinite dimensional Hilbert space, the algebra of bounded operators in the ideal of compact operators, and the quotient map. Throughout this paper A denotes a separable nuclear C*-algebra with unit. An extension of A is a unital *-monomorphism of A into . Two extensions τ1 and τ2 are strongly (weakly) equivalent if there exists a unitary (Fredholm partial isometry) U in such thatfor all a in A.

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.

On the Spectra of Unbounded Subnormal Operators

1986 ◽  
Vol 38 (5) ◽  
pp. 1135-1148 ◽  
Author(s):  
G. McDonald ◽  
C. Sundberg
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Real Line ◽  
Bounded Operator ◽  
Bounded Operators ◽  
The Real ◽  
Subnormal Operators ◽  
Hyponormal Operators ◽  
Image Position ◽  
Topological Boundary

Putnam showed in [5] that the spectrum of the real part of a bounded subnormal operator on a Hilbert space is precisely the projection of the spectrum of the operator onto the real line. (In fact he proved this more generally for bounded hyponormal operators.) We will show that this result can be extended to the class of unbounded subnormal operators with bounded real parts.Before proceeding we establish some notation. If T is a (not necessarily bounded) operator on a Hilbert space, then D(T) will denote its domain, and σ(T) its spectrum. For K a subspace of D(T), T|K will denote the restriction of T to K. Norms of bounded operators and elements in Hilbert spaces will be indicated by ‖ ‖. All Hilbert space inner products will be written 〈,〉. If W is a set in C, the closure of W will be written clos W, the topological boundary will be written bdy W, and the projection of W onto the real line will be written π(W),

10.—An Abstract Relation for Multiparameter Eigenvalue Problems

1976 ◽  
Vol 74 ◽  
pp. 135-143 ◽  
Author(s):  
A. Källström ◽  
B. D. Sleeman
Keyword(s):  
Hilbert Space ◽  
Operator Equation ◽  
Eigenvalue Problems ◽  
Separable Hilbert Space ◽  
Bounded Operator ◽  
Image Position ◽  
Symmetric Operators ◽  
Densely Defined

SynopsisConsider the multiparameter systemwhere ut is an element of a separable Hilbert space Hi, i = 1, …, n. The operators Sij are assumed to be bounded symmetric operators in Hi and Ai is assumed self-adjoint. In addition consider the operator equationwhere B is densely defined and closed in a separable Hilbert space H and Tj, j = 1, …, n is a bounded operator in H. The problem treated in this paper is to seek an expression for a solution v of (**) in terms of the eigenfunctions of the system (*).

Remarks Concerning Uniformly Bounded Operators on Hilbert Space

1972 ◽  
Vol 15 (2) ◽  
pp. 215-217 ◽  
Author(s):  
I. Istrǎțescu
Keyword(s):  
Hilbert Space ◽  
Compact Operator ◽  
Selfadjoint Operator ◽  
Unitary Operator ◽  
Unitary Operators ◽  
Bounded Operators ◽  
Image Position ◽  
Uniformly Bounded

In [6] B. Sz.-Nagy has proved that every operator on a Hilbert space such that1is similar to a unitary operator.The following problem is an extension of this result: If T and S are two operators such that1.sup {‖Tn‖, ‖Sn‖}<∞ (n = 0, ±1, ±2,…)2.TS = STthen there exists a selfadjoint operator Q such that QTQ-1, QSQ-1 are unitary operators?Also, in [7] B. Sz.-Nagy has proved that every compact operator T such thatsup ‖Tn‖<∞ (n = 1, 2, 3,…)is similar to a contraction.

On contractions with spectrum contained in the Cantor set

1995 ◽  
Vol 117 (2) ◽  
pp. 339-343 ◽  
Author(s):  
J. Esterle ◽  
M. Zarrabi ◽  
M. Rajoelina
Keyword(s):  
Hilbert Space ◽  
Cantor Set ◽  
The Other ◽  
Constant Ratio ◽  
Symmetric Set ◽  
Image Position

AbstractFor ξ ε (0, ½) let Eξ be the perfect symmetric set of constant ratio ξ and setIt was shown by the first author that if T is a contraction on the Hilbert space H with spectrum contained in Eξ, and if log ∥T−n∥ = O(nα) as n → ∞ for some α < b(ξ), then T is unitary. In the other direction, we show here that there exists a (non-unitary) contraction T on H such that SpT = Eξ, log ∥T-n∥ = O(nb(ξ)) as n → ∞, and lim supn → ∞ ∥T−n∥ = ∞.

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{&#x0394;} and \delta \in B(B( {\mathcal H} )) denote the elementary operators {\text{&#x0394;}}_{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{&#x0394;} 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{&#x0394;}}_{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.

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