Product Type Operators Involving Radial Derivative Operator Acting between Some Analytic Function Spaces

Let N denote the set of all positive integers and N0=N∪{0}. For m∈N, let Bm={z∈Cm:|z|<1} be the open unit ball in the m−dimensional Euclidean space Cm. Let H(Bm) be the space of all analytic functions on Bm. For an analytic self map ξ=(ξ1,ξ2,…,ξm) on Bm and ϕ1,ϕ2,ϕ3∈H(Bm), we have a product type operator Tϕ1,ϕ2,ϕ3,ξ which is basically a combination of three other operators namely composition operator Cξ, multiplication operator Mϕ and radial derivative operator R. We study the boundedness and compactness of this operator mapping from weighted Bergman–Orlicz space AσΨ into weighted type spaces Hω∞ and Hω,0∞.

Isometric multiplication operators and weighted composition operators of BMOA

Let u be an analytic function in the unit disk $\mathbb{D}$ D and φ be an analytic self-map of $\mathbb{D}$ D . We give characterizations of the symbols u and φ for which the multiplication operator $M_{u}$ M u and the weighted composition operator $M_{u,\varphi }$ M u , φ are isometries of BMOA.

The Quantization of Gravity: Quantization of the Hamilton Equations

We quantize the Hamilton equations instead of the Hamilton condition. The resulting equation has the simple form −Δu=0 in a fiber bundle, where the Laplacian is the Laplacian of the Wheeler–DeWitt metric provided n≠4. Using then separation of variables, the solutions u can be expressed as products of temporal and spatial eigenfunctions, where the spatial eigenfunctions are eigenfunctions of the Laplacian in the symmetric space SL(n,R)/SO(n). Since one can define a Schwartz space and tempered distributions in SL(n,R)/SO(n) as well as a Fourier transform, Fourier quantization can be applied such that the spatial eigenfunctions are transformed to Dirac measures and the spatial Laplacian to a multiplication operator.

The essential norm of multiplication operators acting on Orlicz sequence spaces

We calculate the measure of non-compactness or the essential norm of the multiplication operator Mu acting on Orlicz sequence spaces lφ. As a consequence of our result, we obtain a known criteria for the compactness of multiplication operator acting on lφ.

A note on Nakano generalized difference sequence space

In this paper, we investigate the necessary conditions on any s-type sequence space to form an operator ideal. As a result, we show that the s-type Nakano generalized difference sequence space X fails to generate an operator ideal. We investigate the sufficient conditions on X to be premodular Banach special space of sequences and the constructed prequasi-operator ideal becomes a small, simple, and closed Banach space and has eigenvalues identical with its s-numbers. Finally, we introduce necessary and sufficient conditions on X explaining some topological and geometrical structures of the multiplication operator defined on X.

Eigenvalues of s-type operators on $C(p)$ equipped with a pre-quasi norm

We investigate some new topological properties of the multiplication operator on $C(p)$ C ( p ) defined by Lim (Tamkang J. Math. 8(2):213–220, 1977) equipped with the pre-quasi-norm and the pre-quasi-operator ideal formed by this sequence space and s-numbers.

Noncommutative Functional Calculus and Its Applications on Invariant Subspace and Chaos

Let T:H→H be a bounded linear operator on a separable Hilbert space H. In this paper, we construct an isomorphism Fxx*:L2(σ(|T−a|),μ|T−a|,ξ)→L2(σ(|(T−a)*|),μ|(T−a)*|,Fxx*Hξ) such that (Fxx*)2=identity and Fxx*H is a unitary operator on H associated with Fxx*. With this construction, we obtain a noncommutative functional calculus for the operator T and Fxx*=identity is the special case for normal operators, such that S=R|(S−a)|,ξ(Mzϕ(z)+a)R|S−a|,ξ−1 is the noncommutative functional calculus of a normal operator S, where a∈ρ(T), R|T−a|,ξ:L2(σ(|T−a|),μ|T−a|,ξ)→H is an isomorphism and Mzϕ(z)+a is a multiplication operator on L2(σ(|S−a|),μ|S−a|,ξ). Moreover, by Fxx* we give a sufficient condition to the invariant subspace problem and we present the Lebesgue class BLeb(H)⊂B(H) such that T is Li-Yorke chaotic if and only if T*−1 is for a Lebesgue operator T.