scholarly journals A characterization of the range of a bounded linear transformation in Hilbert space

1980 ◽  
Vol 79 (4) ◽  
pp. 591-591 ◽  
Author(s):  
George O. Golightly
Keyword(s):  
Hilbert Space ◽  
Linear Transformation ◽  
Bounded Linear ◽  
Bounded Linear Transformation

scholarly journals Composition operators on a functional Hilbert space

1979 ◽  
Vol 20 (3) ◽  
pp. 377-384 ◽  
Author(s):  
R.K. Singh ◽  
S.D. Sharma
Keyword(s):  
Hilbert Space ◽  
Lower Bound ◽  
Linear Transformation ◽  
Composition Operator ◽  
Composition Operators ◽  
Bounded Linear ◽  
Functional Hilbert Space ◽  
Bounded Linear Transformation

Let T be a mapping from a set X into itself and let H(X) be a functional Hilbert space on the set X. Then the composition operator CT on H(X) induced by T is a bounded linear transformation from H(X) into itself defined by CTf = f ∘ T. In this paper composition operators are characterized in the case when H(X) = H2(π+) in terms of the behaviour of the inducing functions in the vicinity of the point at infinity. An estimate for the lower bound of ∥CT∥ is given. Also the invertibility of CT is characterized in terms of the invertibility of T.

On operators of class (N, k)

1970 ◽  
Vol 68 (1) ◽  
pp. 141-142 ◽  
Author(s):  
P. B. Ramanujan
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Linear Transformation ◽  
Bounded Linear ◽  
Hyponormal Operators ◽  
Normal Operators ◽  
Bounded Linear Transformation

Istrăţescu (2) has introduced a class of operators on a Banach space called operators of class (N, k). An operator T (a bounded linear transformation) on a Banach space X is said to be an operator of class (N, k), k = 2, 3,…, if ‖Tx‖k ≤ ‖Tkx‖ for all x ∈ X such that ‖x‖ = 1. If k = 2, such an operator is called an operator of class (N)(3). If X is a Hilbert space, then the class of operators of class (N) on X is an extension of the class of hyponormal operators on X (3). The object of this note is to generalize to operators of class (N, k) on a Banach space X, some results which are known to be true for normal operators on a Hilbert space, particularly with regard to their ascent and descent.

Generators of Nest Algebras

1974 ◽  
Vol 26 (3) ◽  
pp. 565-575 ◽  
Author(s):  
W. E. Longstaff
Keyword(s):  
Hilbert Space ◽  
Separable Hilbert Space ◽  
Linear Operators ◽  
Nest Algebra ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Closed Algebra ◽  
Nest Algebras ◽  
Weakly Closed

A collection of subspaces of a Hilbert space is called a nest if it is totally ordered by inclusion. The set of all bounded linear operators leaving invariant each member of a given nest forms a weakly-closed algebra, called a nest algebra. Nest algebras were introduced by J. R. Ringrose in [9]. The present paper is concerned with generating nest algebras as weakly-closed algebras, and in particular with the following question which was first raised by H. Radjavi and P. Rosenthal in [8], viz: Is every nest algebra on a separable Hilbert space generated, as a weakly-closed algebra, by two operators? That the answer to this question is affirmative is proved by first reducing the problem using the main result of [8] and then by using a characterization of nests due to J. A. Erdos [2].

scholarly journals Extensions of symmetric operators I: The inner characteristic function case

2015 ◽  
Vol 2 (1) ◽  
Author(s):  
R.T.W. Martin
Keyword(s):  
Hilbert Space ◽  
Characteristic Function ◽  
Linear Transformation ◽  
Partial Isometry ◽  
Theoretic Characterization ◽  
Symmetric Operators

AbstractGiven a symmetric linear transformation on a Hilbert space, a natural problem to consider is the characterization of its set of symmetric extensions. This problem is equivalent to the study of the partial isometric extensions of a fixed partial isometry. We provide a new function theoretic characterization of the set of all self-adjoint extensions of any symmetric linear transformation B with finite equal indices and inner Livšic characteristic function θ

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.

scholarly journals Pick’s theorems for dissipative operators

Filomat ◽  
2016 ◽  
Vol 30 (6) ◽  
pp. 1591-1599
Author(s):  
H.M. Srivastava ◽  
A.K. Mishra
Keyword(s):  
Hilbert Space ◽  
Analytic Function ◽  
Complex Plane ◽  
Complex Hilbert Space ◽  
Dissipative Operator ◽  
Bounded Linear ◽  
Dissipative Operators ◽  
Proper Contraction ◽  
Complex Valued ◽  
Bounded Linear Transformation

Let H be a complex Hilbert space and let A be a bounded linear transformation on H. For a complex-valued function f, which is analytic in a domain D of the complex plane containing the spectrum of A, let f (A) denote the operator on H defined by means of the Riesz-Dunford integral. In the present paper, several (presumably new) versions of Pick?s theorems are proved for f (A), where A is a dissipative operator (or a proper contraction) and f is a suitable analytic function in the domain D.

scholarly journals A characterization of spectral operators on Hilbert spaces

1982 ◽  
Vol 23 (1) ◽  
pp. 91-95 ◽  
Author(s):  
Ernst Albrecht
Keyword(s):  
Hilbert Space ◽  
Banach Algebra ◽  
Essential Spectrum ◽  
Hilbert Spaces ◽  
Complex Hilbert Space ◽  
Linear Operators ◽  
Spectral Operator ◽  
Bounded Linear ◽  
Image Position

Let H be a complex Hilbert space and denote by B(H) the Banach algebra of all bounded linear operators on H. In [5; 6] J. Ph. Labrousse proved that every operator S∈B(H) which is spectral in the sense of N. Dunford (see [3]) is similar to a T∈B(H) with the following propertyConversely, he showed that given an operator S∈B(H) such that its essential spectrum (in the sense of [5; 6]) consists of at most one point and such that S is similar to a T∈B(H) with the property (1), then S is a spectral operator. This led him to the conjecture that an operator S∈B(H) is spectral if and only if it is similar to a T∈B(H) with property (1). The purpose of this note is to prove this conjecture in the case of operators which are decomposable in the sense of C. Foias (see [2]).

Co-Rank of a Composition Operator

1986 ◽  
Vol 29 (1) ◽  
pp. 33-36 ◽  
Author(s):  
David J. Harrington
Keyword(s):  
Fredholm Operator ◽  
Linear Transformation ◽  
Composition Operator ◽  
Bounded Linear ◽  
Bounded Linear Transformation

AbstractA composition operator CT on L2(X, Σ,m) is a bounded linear transformation induced by a mapping T : X → X via CTf = f∘ T.If m has no atoms then the co-rank of CT (i.e., dim is either zero or infinite. As a corollary, when m has no atoms, CT is a Fredholm operator iff it is invertible.

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