Dedicated to my " father " Leonard Gross, and to the memory of my " grandfather " Irving Segal. Contents 1. Introduction 1 2. Basics of holomorphic function spaces 2 3. Examples of holomorphic function spaces 7 4. A special property of the Segal-Bargmann and weighted Bergman spaces 12 5. Canonical commutation relations 16 6. The Segal-Bargmann transform 21 7. Quantum mechanics and quantization ...

We consider non-selfadjoint operator algebras L(G,λ) generated by weighted creation operators on the Fock-Hilbert spaces of countable directed graphs G. These algebras may be viewed as noncommutative generalizations of weighted Bergman space algebras, or as weighted versions of the free semigroupoid algebras of directed graphs. A complete description of the commutant is obtained together with b...

The boundedness and compactness of weighted composition operators on the Hardy space H of the unit disc is analysed. Particular reference is made to the case when the self-map of the disc is an inner function. Schattenclass membership is also considered; as a result, stronger forms of the two main results of a recent paper of Gunatillake are derived. Finally, weighted composition operators on w...

We classify self-adjoint first-order differential operators on weighted Bergman spaces the unit disc and answer questions related to uncertainty principles for such operators. Our main tools are discrete series representations of $$\textrm{SU}(1,1)$$ . This approach has promise generalize other bounded symmetric domains.

We study composition operators of characteristic zero on weighted Hilbert spaces Dirichlet series. For this purpose, we demonstrate the existence mean counting functions associated with series symbol, and provide a corresponding change variables formula for operator. This leads to natural necessary conditions boundedness compactness. Bergman-type spaces, are able show that compactness condition...

Abstract In this paper, we obtain some interesting reproducing kernel estimates and Carleson properties that play an important role. We characterize the bounded compact Toeplitz operators on weighted Bergman spaces with Békollé-Bonami weights in terms of Berezin transforms. Moreover, estimate essential norm them assuming they are bounded.

We first prove a Cauchy's integral theorem and Cauchy-type formula for certain inhomogeneous Cimmino system from quaternionic analysis perspective. The second part of the paper directs attention towards some applications mentioned results, dealing in particular with four kinds weighted Bergman spaces, reproducing kernels, projection conformal invariant properties.

We describe the boundedness of a linear operator from Bp(ρ) = {f : D → C analytic : (∫ D ρ(1 − |z|) (1 − |z|) |f(z)| dA(z) )1/p < ∞} , for 0 < p ≤ 1 under some conditions on the weight function ρ, into a general Banach space X by means of the growth conditions at the boundary of certain fractional derivatives of a single X-valued analytic function. This, in particular, allows us to characterize...

We use induction and interpolation techniques to prove that a composition operator induced by a map φ is bounded on the weighted Bergman space Aα(H) of the right half-plane if and only if φ fixes ∞ non-tangentially, and has a finite angular derivative λ there. We further prove that in this case the norm, essential norm, and spectral radius of the operator are all equal, and given by λ.

In this note we consider the hyponormality of Toeplitz operators Tφ on the Weighted Bergman space Aα(D) with symbol in the class of functions f + g with polynomials f and g of degree 2. Mathematics subject classification (2010): 47B20, 47B35.