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Register a new userGeneralized Bundle Shift with Application to Multiplication operator on the Bergman space
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Following upon results of Putinar, Sun, Wang, Zheng and the first author, we provide models for the restrictions of the multiplication by a finite Balschke product on the Bergman space in the unit disc to its reducing subspaces. The models involve a generalization of the notion of bundle shift on the Hardy space introduced by Abrahamse and the first author to the Bergman space. We develop generalized bundle shifts on more general domains. While the characterization of the bundle shift is rather explicit, we have not been able to obtain all the earlier results appeared, in particular, the facts that the number of the minimal reducing subspaces equals the number of connected components of the Riemann surface $B(z)=B(w)$ and the algebra of commutant of $T_{B}$ is commutative, are not proved. Moreover, the role of the Riemann surface is not made clear also.
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We study the boundedness and compactness of the generalized Volterra integral operator on weighted Bergman spaces with doubling weights on the unit disk. A generalized Toeplitz operator is defined and the boundedness, compactness and Schatten class of this operator are investigated on the Hilbert weighted Bergman space. As an application, Schatten class membership of generalized Volterra integral operators are also characterized. Finally, we also get the characterizations of Schatten class membership of generalized Toeplitz operator and generalized Volterra integral operators on the Hardy space $H^2$.
We provide a characterization of the commutant of analytic Toeplitz operators $T_B$ induced by finite Blachke products $B$ acting on weighted Bergman spaces which, as a particular instance, yields the case $B(z)=z^n$ on the Bergman space solved recently by by Abkar, Cao and Zhu. Moreover, it extends previous results by Cowen and Wahl in this context and applies to other Banach spaces of analytic functions such as Hardy spaces $H^p$ for $1<p<infty$. Finally, we apply this approach to study reducing subspaces of $T_{B}$ in the classical Bergman space. As a particular instance, we provide a direct proof of a theorem of Hu, Sun, Xu and Yu which states that every analytic Toeplitz operator $T_B$ induced by a finite Blachke product on the Bergman space is reducible and the restriction of $T_B$ on a reducing subspace is unitarily equivalent to the Bergman shift.
In this paper, we study the reducing subspaces for the multiplication operator by a finite Blaschke product $phi$ on the Dirichlet space $D$. We prove that any two distinct nontrivial minimal reducing subspaces of $M_phi$ are orthogonal. When the order $n$ of $phi$ is $2$ or $3$, we show that $M_phi$ is reducible on $D$ if and only if $phi$ is equivalent to $z^n$. When the order of $phi$ is $4$, we determine the reducing subspaces for $M_phi$, and we see that in this case $M_phi$ can be reducible on $D$ when $phi$ is not equivalent to $z^4$. The same phenomenon happens when the order $n$ of $phi$ is not a prime number. Furthermore, we show that $M_phi$ is unitarily equivalent to $M_{z^n} (n > 1)$ on $D$ if and only if $phi = az^n$ for some unimodular constant $a$.
We investigate minimal operator corresponding to operator differential expression with exit from space, study its selfadjoint extensions, also for one particular selfadjoint extension corresponding to boundary value problem with some rational function of eigenparameter in boundary condition establish asymptotics of spectrum and derive trace formula
In this paper we propose a different (and equivalent) norm on $S^{2} ({mathbb{D}})$ which consists of functions whose derivatives are in the Hardy space of unit disk. The reproducing kernel of $S^{2}({mathbb{D}})$ in this norm admits an explicit form, and it is a complete Nevanlinna-Pick kernel. Furthermore, there is a surprising connection of this norm with $3$ -isometries. We then study composition and multiplication operators on this space. Specifically, we obtain an upper bound for the norm of $C_{varphi}$ for a class of composition operators. We completely characterize multiplication operators which are $m$-isometries. As an application of the 3-isometry, we describe the reducing subspaces of $M_{varphi}$ on $S^{2}({mathbb{D}})$ when $varphi$ is a finite Blaschke product of order 2.
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