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 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.

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

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 Compact Weighted Composition Operators and Multiplication Operators between Hardy Spaces

2008 ◽  
Vol 2008 ◽  
pp. 1-12 ◽  
Author(s):  
Sei-Ichiro Ueki ◽  
Luo Luo
Keyword(s):  
Unit Ball ◽  
Composition Operator ◽  
Hardy Spaces ◽  
Composition Operators ◽  
Weighted Composition Operator ◽  
Multiplication Operator ◽  
Essential Norm ◽  
Multiplication Operators ◽  
Weighted Composition Operators ◽  
Weighted Composition

We estimate the essential norm of a compact weighted composition operator acting between different Hardy spaces of the unit ball in . Also we will discuss a compact multiplication operator between Hardy spaces.

scholarly journals Weighted composition operators from Bloch-type into Bers-type spaces

2021 ◽  
Vol 29 (2) ◽  
pp. 243-250
Author(s):  
HAMID VAEZI ◽  
MOHAMAD NAGHLISAR
Keyword(s):  
Composition Operator ◽  
Composition Operators ◽  
Weighted Composition Operator ◽  
Sufficient Conditions ◽  
Weighted Composition Operators ◽  
Type Space ◽  
Bloch Type Space ◽  
Type Spaces ◽  
Weighted Composition ◽  
Necessary And Sufficient

In this paper we consider the weighted composition operator uC_{\varphi} from Bloch-type space B^{\alpha} into Bers-type space H_{\beta}^{\infty}, in three cases, \alpha>1, \alpha=1 and \alpha<1. We give the necessary and sufficient conditions for boundedness and compactness of the above operator.

scholarly journals Spectrums and Uniform Mean Ergodicity of Weighted Composition Operators on Fock Spaces

2021 ◽  
Author(s):  
Werkaferahu Seyoum ◽  
Tesfa Mengestie
Keyword(s):  
Composition Operator ◽  
Composition Operators ◽  
Linear Expansion ◽  
Weighted Composition Operator ◽  
Weighted Composition Operators ◽  
Fock Spaces ◽  
Ergodic Properties ◽  
Weighted Composition ◽  
Mean Ergodicity ◽  
Power Bounded

AbstractFor holomorphic pairs of symbols $$(u, \psi )$$ ( u , ψ ) , we study various structures of the weighted composition operator $$ W_{(u,\psi )} f= u \cdot f(\psi )$$ W ( u , ψ ) f = u · f ( ψ ) defined on the Fock spaces $$\mathcal {F}_p$$ F p . We have identified operators $$W_{(u,\psi )}$$ W ( u , ψ ) that have power-bounded and uniformly mean ergodic properties on the spaces. These properties are described in terms of easy to apply conditions relying on the values |u(0)| and $$|u(\frac{b}{1-a})|$$ | u ( b 1 - a ) | , where a and b are coefficients from linear expansion of the symbol $$\psi $$ ψ . The spectrum of the operators is also determined and applied further to prove results about uniform mean ergodicity.

scholarly journals Supercyclicity and Resolvent Condition for Weighted Composition Operators

2021 ◽  
Author(s):  
Tesfa Mengestie ◽  
Werkaferahu Seyoum
Keyword(s):  
Growth Condition ◽  
Composition Operator ◽  
Composition Operators ◽  
Weighted Composition Operator ◽  
Weighted Composition Operators ◽  
Holomorphic Maps ◽  
Fock Spaces ◽  
Resolvent Condition ◽  
Weighted Composition ◽  
Resolvent Growth

AbstractFor pairs of holomorphic maps $$(u,\psi )$$ ( u , ψ ) on the complex plane, we study some dynamical properties of the weighted composition operator $$W_{(u,\psi )}$$ W ( u , ψ ) on the Fock spaces. We prove that no weighted composition operator on the Fock spaces is supercyclic. Conditions under which the operators satisfy the Ritt’s resolvent growth condition are also identified. In particular, we show that a non-trivial composition operator on the Fock spaces satisfies such a growth condition if and only if it is compact.

scholarly journals WEIGHTED COMPOSITION OPERATORS FROM F(p, q, s) TO BLOCH TYPE SPACES ON THE UNIT BALL

2008 ◽  
Vol 19 (08) ◽  
pp. 899-926 ◽  
Author(s):  
ZE-HUA ZHOU ◽  
REN-YU CHEN
Keyword(s):  
Holomorphic Function ◽  
Unit Ball ◽  
Composition Operator ◽  
Composition Operators ◽  
Bloch Space ◽  
Weighted Composition Operator ◽  
Weighted Composition Operators ◽  
Type Spaces ◽  
Weighted Composition

Let ϕ(z) = (ϕ1(z),…,ϕn(z)) be a holomorphic self-map of B and ψ(z) a holomorphic function on B, where B is the unit ball of ℂn. Let 0 < p, s < +∞, -n - 1 < q < +∞, q+s > -1 and α ≥ 0, this paper characterizes boundedness and compactness of weighted composition operator Wψ,ϕ induced by ϕ and ψ between the space F(p, q, s) and α-Bloch space [Formula: see text].

scholarly journals On the Norm of Certain Weighted Composition Operators on the Hardy Space

2009 ◽  
Vol 2009 ◽  
pp. 1-13 ◽  
Author(s):  
M. Haji Shaabani ◽  
B. Khani Robati
Keyword(s):  
Hardy Space ◽  
Composition Operator ◽  
Composition Operators ◽  
Weighted Composition Operator ◽  
Essential Norm ◽  
Weighted Composition Operators ◽  
Weighted Composition ◽  
Special Case

We obtain a representation for the norm of certain compact weighted composition operator on the Hardy space , whenever and . We also estimate the norm and essential norm of a class of noncompact weighted composition operators under certain conditions on and . Moreover, we characterize the norm and essential norm of such operators in a special case.

Sign in / Sign up

Export Citation Format

Share Document