We investigate the conditions of equivalence of a differential operator of infinite order with constant coefficients to the operator of differentiation in one space of analytic functions. We also study the conditions of continuity of a differential operator of infinite order with variable coefficients in such space.

Characterization of the mapping properties such as boundedness, compactness, measure of non-compactness and estimates of the approximation numbers of Hardy-type integral operators in Banach function spaces are given.

A new version of the maximum principle is presented. The classical Kantorovich-Rubinstein principle gives necessary conditions for the maxima of a linear functional acting on the space of Lipschitzian functions. The maximum value of this functional defines the Hutchinson metric on the space of probability measures. We show an analogous result for the Fortet-Mourier metric. This principle is then applied in the stability theory of Markov-Feller semigroups.

We consider the Fejér (or first arithmetic) means of double Fourier series of functions belonging to one of the Hardy spaces $H^{(1,0)}{(}^2)$, $H^{(0,1)}{(}^2)$, or $H^{(1,1)}{(}^2)$. We prove that the maximal Fejér operator is bounded from $H^{(1,0)}{(}^2)$ or $H^{(0,1)}{(}^2)$ into weak-$L^1{(}^2)$, and also bounded from $H^{(1,1)}{(}^2)$ into $L^1{(}^2)$. These results extend those by Jessen, Marcinkiewicz, and Zygmund, which involve the function spaces $L^{1}lo{g}^{+}L{(}^2)$, $L^1{\left(lo{g}^{+}L\right)}^2{(}^2)$, and $L^{\mu }{(}^2)$ with 0 < μ < 1, respectively. We establish analogous results for the maximal conjugate Fejér operators. On closing, we formulate two conjectures....

We consider Hilbert spaces of analytic functions on a plane domain Ω and multiplication operators on such spaces induced by functions from $H^{\infty }\left(\Omega \right)$. Recently, K. Zhu has given conditions under which the adjoints of multiplication operators on Hilbert spaces of analytic functions belong to the Cowen-Douglas classes. In this paper, we provide some sufficient conditions which give the converse of the main result obtained by K. Zhu. We also characterize the commutant of certain multiplication operators.

Let $\Gamma$ be a rectifiable Jordan curve in the finite complex plane $ℂ$ which is regular in the sense of Ahlfors and David. Denote by $L_C^2\left(\Gamma \right)$ the space of all complex-valued functions on $\Gamma$ which are square integrable w.r. to the arc-length on $\Gamma$. Let $L^2\left(\Gamma \right)$ stand for the space of all real-valued functions in $L_C^2\left(\Gamma \right)$ and put $$L_0^2\left(\Gamma \right)=\{h\in L^2\left(\Gamma \right){\int }_{\Gamma }h\left(\zeta \right)\left|\mathrm{d}\zeta \right|=0\}.$$ Since the Cauchy singular operator is bounded on $L_C^2\left(\Gamma \right)$, the Neumann-Poincaré operator $C_1^{\Gamma }$ sending each $h\in L^2\left(\Gamma \right)$ into $$C_1^{\Gamma }h\left({\zeta }_0\right):=\Re {\left(\pi \mathrm{i}\right)}^{-1}\mathrm{P}.V.{\int }_{\Gamma }\frac{h\left(\zeta \right)}{\zeta -{\zeta }_0}\mathrm{d}\zeta ,{\zeta }_0\in \Gamma ,$$ is bounded on $L^2\left(\Gamma \right)$. We show that the inclusion $$C_1^{\Gamma }\left(L_0^2\left(\Gamma \right)\right)\subset L_0^2\left(\Gamma \right)$$ characterizes the circle in the class of all...

We calculate the spectral multiplicity of the direct sum T⊕ A of a weighted shift operator T on a Banach space Y which is continuously embedded in $l^p$ and a suitable bounded linear operator A on a Banach space X.