scholarly journals Compact composition operators

1979 ◽  
Vol 28 (3) ◽  
pp. 309-314 ◽  
Author(s):  
R. K. Singh ◽  
Ashok Kumar
Keyword(s):  
Composition Operator ◽  
Measure Space ◽  
Composition Operators ◽  
Bounded Operator ◽  
Finite Measure ◽  
Measurable Transformation ◽  
Finite Measure Space

AbstractLet (Хζ,λ) be a σ-finite measure space, and let ϕ be a non-singular measurable transformation from X into itself. Then a composition transformation Cϕ on L2(λ) is defined by Cϕf = f ∘ ϕ. If Cϕ is a bounded operator, then it is called a composition operator. The space L2(λ) is said to admit compact composition operators if there exists a ϕ such that Cϕ is compact. This note is a report on the spaces which admit or which do not admit compact composition operators.

scholarly journals M-Cohyponormal powers of composition operators

1992 ◽  
Vol 53 (1) ◽  
pp. 9-16
Author(s):  
Satish K. Khurana ◽  
Babu Ram
Keyword(s):  
Composition Operator ◽  
Measure Space ◽  
Composition Operators ◽  
Finite Measure ◽  
Finite Measure Space ◽  
Positive Integers ◽  
Nikodym Derivative

AbstractLet T1, i = 1, 2 be measurable transformations which define bounded composition operators C Ti on L2 of a σ-finite measure space. Let us denote the Radon-Nikodym derivative of with respect to m by hi, i = 1, 2. The main result of this paper is that if and are both M-hyponormal with h1 ≤ M2(h2 o T2) a.e. and h2 ≤ M2(h1 o T1) a.e., then for all positive integers m, n and p, []* is -hyponormal. As a consequence, we see that if is an M-hyponormal composition operator, then is -hyponormal for all positive integers n.

scholarly journals Subnormal Weighted Shifts on Directed Trees and Composition Operators inL2-Spaces with Nondensely Defined Powers

2014 ◽  
Vol 2014 ◽  
pp. 1-6 ◽  
Author(s):  
Piotr Budzyński ◽  
Piotr Dymek ◽  
Zenon Jan Jabłoński ◽  
Jan Stochel
Keyword(s):  
Positive Integer ◽  
Composition Operator ◽  
Measure Space ◽  
Composition Operators ◽  
Finite Measure ◽  
Weighted Shift ◽  
Weighted Shifts ◽  
Directed Tree ◽  
Finite Measure Space ◽  
Densely Defined

It is shown that for every positive integernthere exists a subnormal weighted shift on a directed tree (with or without root) whosenth power is densely defined while its (n+1)th power is not. As a consequence, for every positive integernthere exists a nonsymmetric subnormal composition operatorCin anL2-space over aσ-finite measure space such thatCnis densely defined andCn+1is not.

Localising and seminormal composition operators on L2

1994 ◽  
Vol 124 (2) ◽  
pp. 301-316 ◽  
Author(s):  
James T. Campbell ◽  
William E. Hornor
Keyword(s):  
Composition Operator ◽  
Measure Space ◽  
Composition Operators ◽  
Weighted Composition Operator ◽  
Finite Measure ◽  
Reducing Subspace ◽  
Conditional Expectations ◽  
Finite Measure Space ◽  
Weighted Composition

Let (X, ∑, μ) denote a σ-finite measure space. We show that the kernel condition on a weighted composition operator acting on L2(X, ∑, μ), which is necessary for hyponormality of the adjoint, implies that a certain subset of X has the localising property defined by Lambert. For operators satisfying this condition, we find a reducing subspace whose orthocomplement in L2 is annihilated by both the operator and its adjoint, allowing us to obtain characterisations of seminormality for the operator by looking only at the restriction to the reducing subspace. This simplifies the analysis significantly, giving transparent characterisations for the hyponormality and quasinormality of the adjoint, as well as a characterisation of normality for the operator which does not require the computation of any conditional expectations. Several examples are given. We then characterise the semi-hyponormal class for both the operator and its adjoint.

scholarly journals On the almost everywhere convergence of the ergodic averages

1990 ◽  
Vol 10 (1) ◽  
pp. 141-149
Author(s):  
F. J. Martín-Reyes ◽  
A. De La Torre
Keyword(s):  
Measure Space ◽  
Uniform Boundedness ◽  
Finite Measure ◽  
Necessary And Sufficient Condition ◽  
Ergodic Averages ◽  
Measurable Transformation ◽  
Almost Everywhere ◽  
Finite Measure Space ◽  
Finite Invariant Measure ◽  
Necessary And Sufficient

AbstractLet (X, ν) be a finite measure space and let T: X → X be a measurable transformation. In this paper we prove that the averages converge a.e. for every f in Lp(dν), 1 < p < ∞, if and only if there exists a measure γ equivalent to ν such that the averages apply uniformly Lp(dν) into weak-Lp(dγ). As a corollary, we get that uniform boundedness of the averages in Lp(dν) implies a.e. convergence of the averages (a result recently obtained by Assani). In order to do this, we first study measures v equivalent to a finite invariant measure μ, and we prove that supn≥0An(dν/dμ)−1/(p−1) a.e. is a necessary and sufficient condition for the averages to converge a.e. for every f in Lp(dν).

scholarly journals On some classes of weighted composition operators

1990 ◽  
Vol 32 (1) ◽  
pp. 87-94 ◽  
Author(s):  
James T. Campbell ◽  
James E. Jamison
Keyword(s):  
Measure Space ◽  
Composition Operators ◽  
Finite Measure ◽  
Point Transformation ◽  
Weighted Composition Operators ◽  
Absolutely Continuous ◽  
Finite Measure Space ◽  
Weighted Composition ◽  
Image Position ◽  
Complex Valued

Let (X, Σμ) denote a complete a-finite measure space and T: X → X a measurable (T-1A ε Σ each A ε Σ) point transformation from X into itself with the property that the measure μ°T-1 is absolutely continuous with respect to μ. Given any measurable, complex-valued function w(x) on X, and a function f in L2(μ), define WTf(x) via the equation

On the Integrability of the Ergodic Hilbert Transform for A Class of Functions with Equal Absolute Values

1998 ◽  
Vol 5 (2) ◽  
pp. 101-106
Author(s):  
L. Ephremidze
Keyword(s):  
Hilbert Transform ◽  
Measure Space ◽  
Integrable Function ◽  
Finite Measure ◽  
Ergodic Transformation ◽  
Finite Measure Space ◽  
Class Of Functions

Abstract It is proved that for an arbitrary non-atomic finite measure space with a measure-preserving ergodic transformation there exists an integrable function f such that the ergodic Hilbert transform of any function equal in absolute values to f is non-integrable.

Lacunary ideal convergence in measure for sequences of fuzzy valued functions

2021 ◽  
Vol 40 (3) ◽  
pp. 5517-5526
Author(s):  
Ömer Kişi
Keyword(s):  
Measure Space ◽  
Finite Measure ◽  
Ideal Convergence ◽  
Convergence In Measure ◽  
Ideal Form ◽  
Measurable Functions ◽  
Finite Measure Space ◽  
Convergence Of Sequences

We investigate the concepts of pointwise and uniform I θ -convergence and type of convergence lying between mentioned convergence methods, that is, equi-ideally lacunary convergence of sequences of fuzzy valued functions and acquire several results. We give the lacunary ideal form of Egorov’s theorem for sequences of fuzzy valued measurable functions defined on a finite measure space ( X , M , μ ) . We also introduce the concept of I θ -convergence in measure for sequences of fuzzy valued functions and proved some significant results.

On Unsymmetric Dirichlet Forms

1973 ◽  
Vol 25 (2) ◽  
pp. 252-260 ◽  
Author(s):  
Joanne Elliott
Keyword(s):  
Compact Space ◽  
Measure Space ◽  
Closed Subspace ◽  
Finite Measure ◽  
Dirichlet Forms ◽  
Locally Compact ◽  
Locally Compact Space ◽  
Finite Measure Space

Let F be a linear, but not necessarily closed, subspace of L2[X, dm], where (X,,m) is a σ-finite measure space with the Borel subsets of the locally compact space X. If u and v are measureable functions, then v is called a normalized contraction of u if and Assume that F is stable under normalized contractions, that is, if u ∈ F and v is a normalized contraction of u, then v ∈ F.

A Packing Problem for Measurable Sets

1967 ◽  
Vol 19 ◽  
pp. 749-756
Author(s):  
D. Sankoff ◽  
D. A. Dawson
Keyword(s):  
Probability Measure ◽  
Measure Space ◽  
Packing Problem ◽  
Finite Measure ◽  
Finite Measure Space ◽  
Probability Measure Space

Given a probability measure space (Ω,,P)consider the followingpacking problem.What is the maximum number,b(K,Λ), of sets which may be chosen fromso that each set has measureKand no two sets have intersection of measure larger than Λ <K?In this paper the packing problem is solved for any non-atomic probability measure space. Rather than obtaining the solution explicitly, however, it is convenient to solve the followingminimal paving problem.In a non-atomic a-finite measure space (Ω,,μ)what is the measure,V(b, K,Λ), of the smallest set which is the union of exactlybsubsets of measureKsuch that no subsets have intersection of measure larger than Λ?

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