2
votes
1answer
15 views

Characterization of multipliers on the Dirichlet space of holomorphic functions

Can someone tell me whether there is a characterization of boundedness of multiplication operators on the Dirichlet space of the unit disc? These are holomorphic functions $f$ on the unit disk $D$ ...
2
votes
2answers
92 views

How to show that $e^{tA}=\frac{1}{2\pi i}\int_{\{Re \ \lambda =a\}}e^{\lambda t}(\lambda I-A)^{-1}d\lambda$?

Let $X$ be a Banach space and $A:X\to X$ be a bounded operator. We can show that if $|\lambda|>|A|$ then $\lambda I-A$ is invertible and $$(\lambda ...
2
votes
0answers
46 views

Sufficient condition for an operator to be compact in Hilbert space of holomorphic function with respect to Gaussion weight (Fock space).

What I read in a book I could not understand, some one please help. Let $\mathcal{F}=\{f:\mathbb{C^n}\rightarrow\mathbb{C}: \text{$f$ is holomorphic and}\int_{\mathbb{C}^n}\lvert ...
2
votes
1answer
26 views

Definition and analyticity of $T^z$ where $T$ is a positive operator

Let $H$ be a Hilbert space. Suppose that $T\colon D(T) \to H$ is a positive selfadjoint operator where $D(T)$ is the domain of $T$. The spectrum $\sigma(T)$ of the operator $T$ is a subset of ...
0
votes
0answers
47 views

Spectrum of a bounded operator and Liouville's theorem

Let $X$ be a Banach space and $A:X\to X$ be a bounded operator. We know that the spectrum of $A$ is not empty, because otherwise we find a contradiction by using the holomorphy of the function ...
2
votes
0answers
53 views

$\oint_{C}(A-\lambda I)^{-1}\,d\lambda=0$ implies interior of $C$ is in the resolvent.

Suppose that $A$ is a bounded linear operator on a complex Banach space $X$ with resolvent set $\rho(A)$. If $C$ is a simple closed smooth curve in $\rho(A)$ such that $$ ...
3
votes
1answer
147 views

Why study Bergman Spaces?

I'm interested in Operator Algebras and mathematical physics; recently, a friend showed me Duren and Schuster's "Bergman Spaces". I've read about two chapters now and I see there is a nice play ...
2
votes
1answer
103 views

Behavior of the resolvent near the boundary of the spectrum

My question is, in some sense, a continuation of the question below. Isolated singularities of the resolvent Suppose $T\in B(H)$ has no eigenvalues, pick $x\in H$, $x\neq 0$, and consider the ...
3
votes
2answers
94 views

Isolated singularities of the resolvent

Let $T$ be a bounded operator on $l_2$ such that there exists $\mu$ in the spectrum of $T$ which is an isolated point of the spectrum. We know that for any $x\in l_2$ the resolvent function ...
5
votes
1answer
93 views

Adjoint of multiplication by $z$ in the Bergman space

I am learning Hilbert space theory from Halmos' "Introduction to Hilbert space and the theory of spectral multiplicity". While talking about understanding adjoints (p. 39), he calls special ...
1
vote
1answer
29 views

Summation of the Bergman kernel at two distinct points is constant?

Let $\Omega$ be a bounded simply connected domain in $\mathbb{C}.$ Let $K(z,w)$ denotes the Bergman kernel of $\Omega.$ Let $w_1,\,w_2$ be two distinct points in $\Omega.$ I'm looking for a domain ...
3
votes
1answer
35 views

A question on a sequence in a Banach algebra [duplicate]

If $\{u_{k}\}_{k=1}^{\infty}$ is a sequence in an Banach algebra (and more specifically, in the set of all the bounded linear operators of a Banach space $X$). If ...
2
votes
2answers
90 views

Show that the operator is bounded in $L_p$

Consider the operator $C$, acting on functions $f$ on the unit circle $S^1 = \left\{ z \in \mathbb C \mid |z| = 1 \right\}$ by the rule $$ (Cf)(z) = \frac{1}{2\pi i} ...
1
vote
0answers
35 views

Composition of analytic functions is analytic in a general setting, and are they continuous?

Regarding the notion of analyticity discussed in this setting: A possible equivalence for holomorphicity I wonder if this is truly the correct definition (even though it is from Dunford-Schwarz) An ...
0
votes
0answers
26 views

A possible equivalence for holomorphicity

Let $X$ and $Y$ be Complex Banach spaces with $U$ an open subset of $X$, and $f:U\rightarrow Y$. We say that f is analytic/holomorphic if for every $z_1, ... z_n \in X$ we have that the mapping $a_1, ...
1
vote
0answers
53 views

Almost everywhere analytic function

Suppose we have a measure space $\Omega$ and a function $m\in L^\infty(\Omega,\mathcal{B}(E))$, that is invertible for almost all $\theta\in\Omega$ Further assume, that we have an other function $G$ ...
3
votes
1answer
37 views

Extension of family of operators

Let $A(z)$ where $z\in \mathbb{R}$ be a family of (bounded) operators on some Hilbert space. Assume we know these operators have a meromorphic extension to all of $\mathbb{C}$. Assume moreover that we ...
1
vote
1answer
74 views

Composition of $\mathrm H^p$ function with Möbius transform

Let $f:\mathbb D\rightarrow \mathbb C$ be a function in $\mathrm{H}^p$, i.e. $$\exists M>0,\text{ such that }\int_0^{2\pi}|f(re^{it})|^pdt\leq M<\infty,\forall r\in[o,1)$$ Consider a Möbius ...
1
vote
1answer
58 views

Showing a bound on a contour integral

I'm working through M. Schechter's 'Principles of Functional Analysis' and I'm working through a proof on page 136 that shows that the spectral radius $r_{\sigma} (T) $ of a bounded linear operator ...
7
votes
1answer
207 views

When are two commuting linear operators functions of each other

I've computed that the following is valid for certain functions but I've hit a slight bump in my proof. I was wondering if someone could clear it up. If we formally consider the integral operators ...
2
votes
0answers
132 views

winding number question

This is part of a proof from Banach algebra techniques in Operator theory by Ronald Douglas on page 170. Let $\epsilon>0$. Let $T$ be the unit circle and $\phi\in H^\infty+C(T)$. Choose $\psi\in ...
2
votes
0answers
84 views

eigenvalue question for a Toeplitz operator

Let $\phi$ be a nonzero function in $L^\infty(T)$ where $T$ is the unit circle. Let $M_\phi$ be the multiplication operator and $T_\phi$ be the Toeplitz operator. Show $T_\phi$ and $M_\phi$ have no ...
6
votes
1answer
134 views

invariant subspace of a Hardy space

Let $T$ be the unit circle and $H^1=\{f\in L^1(T): \int_0^{2\pi} f(e^{it})\chi_n(e^{it})dt=0 \text{ for } n>0\}$ where $\chi_n(e^{it})=e^{int}$. Let $M$ be a closed subspace of $H^1$. Then ...
6
votes
0answers
295 views

Hilbert transform and Hilbert matrix

The Hilbert matrix is \begin{bmatrix} 1 & \frac{1}{2} & \frac{1}{3} & \frac{1}{4} & \dots \\[4pt] \frac{1}{2} & \frac{1}{3} & \frac{1}{4} & & \ddots \\[4pt] ...
2
votes
1answer
58 views

Injectivity of a certain operator

Consider a compact $K$ of the complex plane of interior void, $f$ a rational fraction leaving $K$ invariant (i.e. $f(K) \subset K$), and $\mu$ a borelian probability measure supported by $K$, and ...
4
votes
0answers
371 views

Confused by a proof in Rudin *Functional Analysis*

I am reading Rudin's Functional Analysis and got quite confused by his proof of Thm 8.5, that is, the existence of fundamental solutions for differential operator $P(D)$, where $P$ is a polynomial. ...
0
votes
1answer
112 views

Weak analyticity vs. Strong Analyticity

Let $X$ be a (complex) banach space, $U$ be an open subset of $\mathbb{C}$ and $f: U \to X$ be a function that is completely arbitrary except that it satisfies the property that for any continuous ...
6
votes
1answer
349 views

Lie group heuristics for a raising operator for $(-1)^n \frac{d^n}{d\beta^n}\frac{x^\beta}{\beta!}|_{\beta=0}$

Consider the fractional integro-derivative $\displaystyle\frac{d^{\beta}}{dx^\beta}\frac{x^{\alpha}}{\alpha!}=FP\frac{1}{2\pi ...
2
votes
0answers
234 views

Cauchy's integral formula for operators

I study this article : A supersymmetric transfer matrix and differentiability of the density of states in the one-dimensional Anderson model. Massimo Campanino and Abel Klein. Comm. Math. Phys. 104 ...
2
votes
1answer
251 views

Simple isolated eigenvalue and pole of the resolvent

Let $T$ be bounded linear operator on some complex Banach space, and $\lambda$ an eigenvalue of $T$ which is isolated in its spectrum, and such that $\bigcup_{n\ge 1} N((T- \lambda I)^n)$ is ...