In this note we study sub-Hardy Hilbert spaces on which the the action of the operator of multiplication by the coordinate function z is assumed to be weaker than that of an isometry. We identify such operators with a class of weighted shifts. The well known results of de Branges and Beurling are deduced as corollaries .
Let $mathcal{D}$ be the class of radial weights on the unit disk which satisfy both forward and reverse doubling conditions. Let $g$ be an analytic function on the unit disk $mathbb{D}$. We characterize bounded and compact Volterra type integration o
perators [ J_{g}(f)(z)=int_{0}^{z}f(lambda)g(lambda)dlambda ] between weighted Bergman spaces $L_{a}^{p}(omega )$ induced by $mathcal{D}$ weights and Hardy spaces $H^{q}$ for $0<p,q<infty$.
The possibility of defining sesquilinear forms starting from one or two sequences of elements of a Hilbert space is investigated. One can associate operators to these forms and in particular look for conditions to apply representation theorems of ses
quilinear forms, such as Katos theorems. The associated operators correspond to classical frame operators or weakly-defined multipliers in the bounded context. In general some properties of them, such as the invertibility and the resolvent set, are related to properties of the sesquilinear forms. As an upshot of this approach new features of sequences (or pairs of sequences) which are semi-frames (or reproducing pairs) are obtained.
We characterize the (essentially) decreasing sequences of positive numbers $beta$ = ($beta$ n) for which all composition operators on H 2 ($beta$) are bounded, where H 2 ($beta$) is the space of analytic functions f in the unit disk such that $infty$
n=0 |c n | 2 $beta$ n < $infty$ if f (z) = $infty$ n=0 c n z n. We also give conditions for the boundedness when $beta$ is not assumed essentially decreasing.
Let $X$ be a space of homogeneous type and $L$ be a nonnegative self-adjoint operator on $L^2(X)$ satisfying Gaussian upper bounds on its heat kernels. In this paper we develop the theory of weighted Besov spaces $dot{B}^{alpha,L}_{p,q,w}(X)$ and wei
ghted Triebel--Lizorkin spaces $dot{F}^{alpha,L}_{p,q,w}(X)$ associated to the operator $L$ for the full range $0<p,qle infty$, $alphain mathbb R$ and $w$ being in the Muckenhoupt weight class $A_infty$. Similarly to the classical case in the Euclidean setting, we prove that our new spaces satisfy important features such as continuous charaterizations in terms of square functions, atomic decompositions and the identifications with some well known function spaces such as Hardy type spaces and Sobolev type spaces. Moreover, with extra assumptions on the operator $L$, we prove that the new function spaces associated to $L$ coincide with the classical function spaces. Finally we apply our results to prove the boundedness of the fractional power of $L$ and the spectral multiplier of $L$ in our new function spaces.
We completely characterize the simultaneous membership in the Schatten ideals $S_ p$, $0<p<infty$ of the Hankel operators $H_ f$ and $H_{bar{f}}$ on the Bergman space, in terms of the behaviour of a local mean oscillation function, proving a conjecture of Kehe Zhu from 1991.