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.

Let ϕ: [0,1] → [0,1] be a nondecreasing continuous function such that ϕ(x) > x for all x ∈ (0,1). Let the operator $V_{\varphi }:f\left(x\right)\mapsto {\int }_0^{\varphi \left(x\right)}f\left(t\right)dt$ be defined on L₂[0,1]. We prove that $V_{\varphi }$ has a finite number of nonzero eigenvalues if and only if ϕ(0) > 0 and ϕ(1-ε) = 1 for some 0 < ε < 1. Also, we show that the spectral trace of the operator $V_{\varphi }$ always equals 1.

We present a proof of the weighted estimate of the Bergman projection that does not use extrapolation results. This estimate is extended to product domains using an adapted definition of Békollé-Bonami weights in this setting. An application to bounded Toeplitz products is also given.

In this paper we study some properties of a totally $*$-paranormal operator (defined below) on Hilbert space. In particular, we characterize a totally $*$-paranormal operator. Also we show that Weyl’s theorem and the spectral mapping theorem hold for totally $*$-paranormal operators through the local spectral theory. Finally, we show that every totally $*$-paranormal operator satisfies an analogue of the single valued extension property for $W^2(D,H)$ and some of totally $*$-paranormal operators have scalar extensions....

We show that a stochastic operator acting on the Banach lattice $L^1\left(m\right)$ of all $m$-integrable functions on $(X,𝒜)$ is quasi-compact if and only if it is uniformly smoothing (see the definition below).

Let $\varphi$ be an analytic self-mapping of $𝔻$ and $g$ an analytic function on $𝔻$. In this paper we characterize the bounded and compact Volterra composition operators from the Bergman-type space to the Bloch-type space. We also obtain an asymptotical expression of the essential norm of these operators in terms of the symbols $g$ and $\varphi$.