Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userSpectral theory of multiplication operators on Hardy-Sobolev spaces
152
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
For a pointwise multiplier $varphi$ of the Hardy-Sobolev space $H^2_beta$ on the open unit ball $bn$ in $cn$, we study spectral properties of the multiplication operator $M_varphi: H^2_betato H^2_beta$. In particular, we compute the spectrum and essential spectrum of $M_varphi$ and develop the Fredholm theory for these operators.
rate research
Read More
For any real $beta$ let $H^2_beta$ be the Hardy-Sobolev space on the unit disk $D$. $H^2_beta$ is a reproducing kernel Hilbert space and its reproducing kernel is bounded when $beta>1/2$. In this paper, we study composition operators $C_varphi$ on $H^2_beta$ for $1/2<beta<1$. Our main result is that, for a non-constant analytic function $varphi:DtoD$, the operator $C_{varphi }$ has dense range in $H_{beta }^{2}$ if and only if the polynomials are dense in a certain Dirichlet space of the domain $varphi(D)$. It follows that if the range of $C_{varphi }$ is dense in $H_{beta }^{2}$, then $varphi $ is a weak-star generator of $H^{infty}$. Note that this conclusion is false for the classical Dirichlet space $mathfrak{D}$. We also characterize Fredholm composition operators on $H^{2}_{beta }$.
In this paper, we investigate the boundedness of Toeplitz product $T_{f}T_{g}$ and Hankel product $H_{f}^{*} H_{g}$ on Fock-Sobolev space for two polynomials $f$ and $g$ in $z,overline{z}inmathbb{C}^{n}$. As a result, the boundedness of Toeplitz operator $T_{f}$ and Hankel operator $H_{f}$ with the polynomial symbol $f$ in $z,overline{z}inmathbb{C}^{n}$ is characterized.
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.
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 $mathcal{D}$ 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 operators [ 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 $0<p,q<infty$.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
