scholarly journals On (m, P)-expansive operators: products, perturbation by nilpotents, Drazin invertibility

2021 ◽  
Vol 8 (1) ◽  
pp. 158-175
Author(s):  
B.P. Duggal
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Multiplication Operators ◽  
Sufficient Condition ◽  
Hilbert Space Operators ◽  
Algebraic Properties ◽  
Left And Right ◽  
Banach Space Operators

Abstract A generalisation of m-expansive Hilbert space operators T ∈ B(ℋ) [18, 20] to Banach space operators T ∈ B(𝒳) is obtained by defining that a pair of operators A, B ∈ B(𝒳) is (m, P)-expansive for some operator P ∈ B(𝒳) if Δ A,B m (P)= ( I - L A R B ) m ( P ) = ∑ j = 0 m ( - 1 ) j ( j m ) {\left( {I - {L_A}{R_B}} \right)^m}\left( P \right) = \sum\nolimits_{j = 0}^m {{{\left( { - 1} \right)}^j}\left( {_j^m} \right)} AjPBj ≤0; LA(X) = AX and RB(X)=XB. Unlike m-isometric and m-left invertible operators, commuting products and perturbations by commuting nilpotents of (m, I)-expansive operators do not result in expansive operators: using elementary algebraic properties of the left and right multiplication operators, a sufficient condition is proved. For Drazin invertible A and B ∈ B(ℋ), with Drazin inverses Ad and Bd, a sufficient condition proving (Ad, Bd) ^ (A, B) is (m − 1, P)-isometric (resp., (m − 1, P)-contractive) for m even (resp., m odd) is given, and a Banach space analogue of this result is proved.

scholarly journals On the largest analytic set for cyclic operators

2003 ◽  
Vol 2003 (30) ◽  
pp. 1899-1909
Author(s):  
A. Bourhim
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Bounded Linear ◽  
Analytic Set ◽  
Point Evaluations ◽  
Analytic Bounded Point Evaluations ◽  
Cyclic Operators ◽  
Banach Space Operators

We describe the set of analytic bounded point evaluations for an arbitrary cyclic bounded linear operatorTon a Hilbert spaceℋ; some related consequences are discussed. Furthermore, we show that two densely similar cyclic Banach-space operators possessing Bishop's property(β)have equal approximate point spectra.

scholarly journals Banach space operators with a bounded H∞ functional calculus

1996 ◽  
Vol 60 (1) ◽  
pp. 51-89 ◽  
Author(s):  
Michael Cowling ◽  
Ian Doust ◽  
Alan Micintosh ◽  
Atsushi Yagi
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Linear Operator ◽  
Functional Calculus ◽  
General Definition ◽  
Square Function ◽  
Square Function Estimates ◽  
Imaginary Powers ◽  
Banach Space Operators ◽  
General Banach Space

AbstractIn this paper, we give a general definition for f(T) when T is a linear operator acting in a Banach space, whose spectrum lies within some sector, and which satisfies certain resolvent bounds, and when f is holomorphic on a larger sector.We also examine how certain properties of this functional calculus, such as the existence of a bounded H∈ functional calculus, bounds on the imaginary powers, and square function estimates are related. In particular we show that, if T is acting in a reflexive Lp space, then T has a bounded H∈ functional calculus if and only if both T and its dual satisfy square function estimates. Examples are given to show that some of the theorems that hold for operators in a Hilbert space do not extend to the general Banach space setting.

scholarly journals On nth roots of normal operators

Filomat ◽  
2020 ◽  
Vol 34 (8) ◽  
pp. 2797-2803
Author(s):  
B.P. Duggal ◽  
I.H. Kim
Keyword(s):  
Hilbert Space ◽  
Type Theorem ◽  
Sufficient Condition ◽  
Property A ◽  
Class A ◽  
Hilbert Space Operators ◽  
Normal Operators

For n-normal operators A [2,4,5], equivalently n-th roots A of normal Hilbert space operators, both A and A* satisfy the Bishop-Eschmeier-Putinar property (?)?, A is decomposable and the quasinilpotent part H0(A-?) of A satisfies H0(A-?=)-1(0) = (A-?)-1(0) for every non-zero complex ?. A satisfies every Weyl and Browder type theorem, and a sufficient condition for A to be normal is that either A is dominant or A is a class A(1,1) operator.

scholarly journals Some algebraic properties of F(X) and K(X)

1975 ◽  
Vol 19 (4) ◽  
pp. 353-361 ◽  
Author(s):  
Freda E. Alexander
Keyword(s):  
Banach Space ◽  
Finite Rank ◽  
Approximation Property ◽  
Reflexive Banach Space ◽  
Compact Operators ◽  
Multiplication Operators ◽  
Weakly Compact ◽  
Dual Algebra ◽  
Left Multiplication ◽  
Algebraic Properties

Throughout we consider operators on a reflexive Banach space X. We consider certain algebraic properties of F(X), K(X) and B(X) with the general aim of examining their dependence on the possession by X of the approximation property. B(X) (resp. K(X)) denotes the algebra of all bounded (resp. compact) operators on X and F(X) denotes the closure in B(X) of its finite rank operators. The two questions we consider are:(1) Is K(X) equal to the set of all operators in B(X) whose right and left multiplication operators on F(X) (or on B(X)) are weakly compact?(2) Is F(X) a dual algebra?

scholarly journals A New Exceptional Family of Elements and Solvability of General Order Complementarity Problems

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Na Huang ◽  
Changfeng Ma
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Complementarity Problems ◽  
Sufficient Conditions ◽  
Necessary Condition ◽  
Sufficient Condition ◽  
Exceptional Family Of Elements ◽  
Exceptional Family ◽  
General Order ◽  
Sufficient And Necessary Condition

By using the concept of exceptional family, we propose a sufficient condition of a solution to general order complementarity problems (denoted by GOCP) in Banach space, which is weaker than that in Németh, 2010 (Theorem 3.1). Then we study some sufficient conditions for the nonexistence of exceptional family for GOCP in Hilbert space. Moreover, we prove that without exceptional family is a sufficient and necessary condition for the solvability of pseudomonotone general order complementarity problems.

scholarly journals A characterisation of Hilbert spaces via orthogonality and proximinality

2005 ◽  
Vol 71 (1) ◽  
pp. 107-111
Author(s):  
Fathi B. Saidi
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Banach Spaces ◽  
Hilbert Spaces ◽  
Real Banach Space ◽  
Nonzero Element ◽  
Dimensional Subspace ◽  
Two Dimensional ◽  
Orthogonal Direction ◽  
Open Problems

In this paper we adopt the notion of orthogonality in Banach spaces introduced by the author in [6]. There, the author showed that in any two-dimensional subspace F of E, every nonzero element admits at most one orthogonal direction. The problem of existence of such orthogonal direction was not addressed before. Our main purpose in this paper is the investigation of this problem in the case where E is a real Banach space. As a result we obtain a characterisation of Hilbert spaces stating that, if in every two-dimensional subspace F of E every nonzero element admits an orthogonal direction, then E is isometric to a Hilbert space. We conclude by presenting some open problems.

scholarly journals A note on best approximation and invertibility of operators on uniformly convex Banach spaces

1991 ◽  
Vol 14 (3) ◽  
pp. 611-614 ◽  
Author(s):  
James R. Holub
Keyword(s):  
Banach Space ◽  
Linear Operator ◽  
Banach Spaces ◽  
Best Approximation ◽  
Bounded Linear Operator ◽  
Convex Banach Space ◽  
Sufficient Condition ◽  
Uniformly Convex ◽  
Bounded Linear ◽  
Uniformly Convex Banach Spaces

It is shown that ifXis a uniformly convex Banach space andSa bounded linear operator onXfor which‖I−S‖=1, thenSis invertible if and only if‖I−12S‖<1. From this it follows that ifSis invertible onXthen either (i)dist(I,[S])<1, or (ii)0is the unique best approximation toIfrom[S], a natural (partial) converse to the well-known sufficient condition for invertibility thatdist(I,[S])<1.

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.

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