scholarly journals On Convexity of Composition and Multiplication Operators on Weighted Hardy Spaces

2009 ◽  
Vol 2009 ◽  
pp. 1-9 ◽  
Author(s):  
Karim Hedayatian ◽  
Lotfollah Karimi
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Composition Operator ◽  
Hardy Spaces ◽  
Bounded Linear Operator ◽  
Sufficient Conditions ◽  
Multiplication Operators ◽  
Weighted Hardy Spaces ◽  
Bounded Linear ◽  
Necessary And Sufficient

A bounded linear operatorTon a Hilbert spaceℋ, satisfying‖T2h‖2+‖h‖2≥2‖Th‖2for everyh∈ℋ, is called a convex operator. In this paper, we give necessary and sufficient conditions under which a convex composition operator on a large class of weighted Hardy spaces is an isometry. Also, we discuss convexity of multiplication operators.

Products of Normal Operators

1988 ◽  
Vol 40 (6) ◽  
pp. 1322-1330 ◽  
Author(s):  
Pei Yuan Wu
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Separable Hilbert Space ◽  
Polar Decomposition ◽  
Sufficient Conditions ◽  
Bounded Linear ◽  
Underlying Space ◽  
Finite Dimensional ◽  
Normal Operators ◽  
Necessary And Sufficient

Which bounded linear operator on a complex, separable Hilbert space can be expressed as the product of finitely many normal operators? What is the answer if “normal” is replaced by “Hermitian”, “nonnegative” or “positive”? Recall that an operator T is nonnegative (resp. positive) if (Tx, x) ≧ 0 (resp. (Tx, x) ≥ 0) for any x ≠ 0 in the underlying space. The purpose of this paper is to provide complete answers to these questions.If the space is finite-dimensional, then necessary and sufficient conditions for operators expressible as such are already known. For normal operators, this is easy. By the polar decomposition, every operator is the product of two normal operators. An operator is the product of Hermitian operators if and only if its determinant is real; moreover, in this case, 4 Hermitian operators suffice and 4 is the smallest such number (cf. [10]).

scholarly journals Bounded point evaluations for cyclic Hilbert space operators

2003 ◽  
Vol 4 (2) ◽  
pp. 301
Author(s):  
A. Bourhim
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Bounded Linear ◽  
Hilbert Space Operators ◽  
Point Evaluations ◽  
Analytic Bounded Point Evaluations

<p>In this talk, to be given at a conference at Seconda Università degli Studi di Napoli in September 2001, we shall describe the set of analytic bounded point evaluations for an arbitrary cyclic bounded linear operator T on a Hilbert space H and shall answer some questions due to L. R. Williams.</p>

scholarly journals Exponential Function of a bounded Linear Operator on a Hilbert Space.

2014 ◽  
Vol 11 (3) ◽  
pp. 1267-1273
Author(s):  
Baghdad Science Journal
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Exponential Function ◽  
Bounded Linear Operator ◽  
Bounded Linear ◽  
Exponential Operator

In this paper, we introduce an exponential of an operator defined on a Hilbert space H, and we study its properties and find some of properties of T inherited to exponential operator, so we study the spectrum of exponential operator e^T according to the operator T.

The Set of Finite Operators is Nowhere Dense

1989 ◽  
Vol 32 (3) ◽  
pp. 320-326 ◽  
Author(s):  
Domingo A. Herrero
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Dense Subset ◽  
Bounded Linear ◽  
Infinite Dimensional ◽  
Infinite Dimensional Hilbert Space ◽  
Dimensional Hilbert Space

AbstractA bounded linear operator A on a complex, separable, infinite dimensional Hilbert space is called finite if for each . It is shown that the class of all finite operators is a closed nowhere dense subset of

On Self-Adjoint Factorization of Operators

1969 ◽  
Vol 21 ◽  
pp. 1421-1426 ◽  
Author(s):  
Heydar Radjavi
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Unitary Operator ◽  
Normal Operator ◽  
Complex Hilbert Space ◽  
Bounded Linear ◽  
Infinite Dimensional ◽  
Normal Operators ◽  
Adjoint Operators

The main result of this paper is that every normal operator on an infinitedimensional (complex) Hilbert space ℋ is the product of four self-adjoint operators; our Theorem 4 is an actually stronger result. A large class of normal operators will be given which cannot be expressed as the product of three self-adjoint operators.This work was motivated by a well-known resul t of Halmos and Kakutani (3) that every unitary operator on ℋ is the product of four symmetries, i.e., operators that are self-adjoint and unitary.1. By “operator” we shall mean bounded linear operator. The space ℋ will be infinite-dimensional (separable or non-separable) unless otherwise specified. We shall denote the class of self-adjoint operators on ℋ by and that of symmetries by .

scholarly journals Sums of A Pair of Orthogonal Frames

Mathematics ◽  
2019 ◽  
Vol 7 (7) ◽  
pp. 582
Author(s):  
Ghanshyam Bhatt
Keyword(s):  
Linear Operator ◽  
Bounded Linear Operator ◽  
Sufficient Condition ◽  
Simple Construction ◽  
Necessary And Sufficient Condition ◽  
Dual Frames ◽  
Bounded Linear ◽  
Frame Bounds ◽  
Necessary And Sufficient

Frames are more stable as compared to bases under the action of a bounded linear operator. Sums of different frames under the action of a bounded linear operator are studied with the help of analysis, synthesis and frame operators. A simple construction of frames from the existing ones under the action of such an operator is presented here. It is shown that a frame can be added to its alternate dual frames, yielding a new frame. It is also shown that the sum of a pair of orthogonal frames is a frame. This provides an easy construction of a frame where the frame bounds can be computed easily. Moreover, for a pair of orthogonal frames, the necessary and sufficient condition is presented for their alternate dual frames to be orthogonal. This allows for an easy construction of a large number of new frames.

scholarly journals Antinormal composition operators on $ \mbf{\ell^2}$

2008 ◽  
Vol 39 (4) ◽  
pp. 347-352 ◽  
Author(s):  
Gyan Prakash Tripathi ◽  
Nand Lal
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Composition Operators ◽  
Complete Characterization ◽  
Inner Product ◽  
Complex Numbers ◽  
Bounded Linear ◽  
Normal Operators

A bounded linear operator $ T $ on a Hilbert space $ H $ is called antinormal if the distance of $ T $ from the set of all normal operators is equal to norm of $ T $. In this paper, we give a complete characterization of antinormal composition operators on $ \ell^2 $, where $ \ell^2 $ is the Hilbert space of all square summable sequences of complex numbers under standard inner product on it.

Representation of certain Linear Operators in Hilbert Space

1971 ◽  
Vol 23 (1) ◽  
pp. 132-150 ◽  
Author(s):  
Bernard Niel Harvey
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Separable Hilbert Space ◽  
Finite Dimension ◽  
Linear Operators ◽  
Inner Product ◽  
Indefinite Metric ◽  
Bounded Linear ◽  
Bilinear Functional

In this paper we represent certain linear operators in a space with indefinite metric. Such a space may be a pair (H, B), where H is a separable Hilbert space, B is a bilinear functional on H given by B(x, y) = [Jx, y], [, ] is the Hilbert inner product in H, and J is a bounded linear operator such that J = J* and J2 = I. If T is a linear operator in H, then ‖T‖ is the usual operator norm. The operator J above has two eigenspaces corresponding to the eigenvalues + 1 and –1.In case the eigenspace in which J induces a positive operator has finite dimension k, a general spectral theory is known and has been developed principally by Pontrjagin [25], Iohvidov and Kreĭn [13], Naĭmark [20], and others.

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.

Numerical Range and Convex Sets*

1974 ◽  
Vol 17 (2) ◽  
pp. 295-296 ◽  
Author(s):  
Fredric M. Pollack
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Convex Subset ◽  
Bounded Linear Operator ◽  
Convex Sets ◽  
Numerical Range ◽  
Bounded Subset ◽  
Bounded Linear ◽  
Empty Convex ◽  
Image Position

The numerical range W(T) of a bounded linear operator T on a Hilbert space H is defined byW(T) is always a convex subset of the plane [1] and clearly W(T) is bounded since it is contained in the ball of radius ‖T‖ about the origin. Which non-empty convex bounded subsets of the plane are the numerical range of an operator? The theorem we prove below shows that every non-empty convex bounded subset of the plane is W(T) for some T.

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