In the present work, using a formula describing all scalar spectral functions of a Carleman operator $A$ of defect indices $(1,1)$ in the Hilbert space $L^2(X,\mu )$ that we obtained in a previous paper, we derive certain results concerning the localization of the spectrum of quasi-selfadjoint extensions of the operator $A$.

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...

Let A and B be two -non necessarily bounded- normal operators. We give new conditions making their product normal. We also generalize a result by Deutsch et al on normal products of matrices.