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

Abstract

For 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.

Weighted Banach spaces of analytic functions with sup-norms and operators between them: a survey

“…The set of periodic points of these operators is also determined. This research is continued for weighted composition operators in [148].…”

Dynamics of Weighted Composition Operators and Their Adjoints on the Fock Space

“…If it were cyclic, then a classical result of Von Neumann [15] implies that the operator has simple spectrum. On the contrary, by [16,Theorem 3.1], the spectrum of W (u,ψ) is the set…”

“…Since then, the operator has become a natural object of study and its investigation has rapidly evolved in function related operator theory. A number of researchers have studied the operator in various settings expressing its spectral and topological properties in terms of the function theoretic properties of the inducing pairs of symbols (u, ψ); see for example [2,5,11,12,16,17] and the references therein.…”