0
votes
1answer
7 views

Integration over subsets of the complex plane.

Original Problem: Let $\Omega\subset \mathbb{C}$ be an open set and let $f:\Omega\to\mathbb{C}$ be holomorphic such that $f\in L^{2}(\Omega)$. Show that if $B(z,r)$, the ball of radius $r$ ...
1
vote
0answers
59 views

how to prove $\sum_n |b_n|^2<\infty$

$\{b_n\}$ is a complex sequence, If for all $\ell^2$ sequences $a_n$, we have $\sum_n \bar{a}_nb_n$ converges . Prove that $\sum_n|b_n|^2<\infty$
4
votes
1answer
32 views

Isomorphism between Hilbert spaces

I want to show that the function $$ L^2(\Omega,\mathcal{O})\longrightarrow L^2(\widetilde{\Omega},\mathcal{O}) \colon f \longmapsto f|_{\widetilde{\Omega}}$$ is a isomorphism, where ...
1
vote
1answer
32 views

Is every closed set $K\subseteq \mathbb{C}$ the essential range of a measurable function?

For a complex-valued function $h$ on a measure space $(S,\Sigma, \mu)$, the $\textit{essential range}$ of $h$ is the set of all $\lambda \in \mathbb{C}$ such that for all $\epsilon >0$ the ...
4
votes
1answer
222 views

functions orthogonal to the exponential Bell polynomials

Consider the single variable Bell polynomials $\phi_{n}(x)$ given by: $$\phi_{n}(x)=e^{-x}\sum_{k=0}^{\infty}\frac{k^{n}x^{k}}{k!}$$ I am looking for a set of functions $\tilde{\phi}_{n}(x)$ such ...
0
votes
1answer
33 views

I want to show one norm is less than or equal to another norm on C([0,1])

Let $|| \ ||_1$ be the norm on $C([0,1])$ defined by $||f||_1 = \int_0^1|f(t)|dt$. a) Show that $||f||_1 \le ||f||_{[0,1]}$ b) Are $|| \ ||_1$ and $|| \ ||_{[0,1]}$ equivalent? For part a) I think ...
5
votes
1answer
85 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 ...
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
44 views

Prove that $S$is a closed subspace of $H^2$ invariant under multiplication by $z$. Find the inner function $F$ such that $S=FH^2$

Let ${\alpha_n}$ be a sequence of points in the open unit disc such that $\sum(1-|\alpha_n|)<\infty$. Let $S$ be the set of all functions $f$ in $H^2$ spaces such that $f(\alpha_n)=f'(\alpha_n)=0$ ...
1
vote
0answers
50 views

Alternative explanation for $\iint_D \left|\log \left( \frac{e}{1-z} \right) \right|^2 \ dA = \frac{\pi^3}{6}$?

I thought up a curious definite integral. Let $D = \{ z \in \mathbb{C} : |z|<1\}$. Let $A$ denote area measure on $D$, normalized so that $A(D) = \pi$. I claim that $$\iint_D \left|\log \left( ...
6
votes
3answers
309 views

Why isn't it a Hilbert space

Let $X$ be the vector space of all the continuous complex-valued functions on $[0,1]$. Then $X$ has an inner product $$(f,g) = \int_0^1 f(t)\overline{g(t)} dt$$ to make it an inner product space. But ...
8
votes
0answers
348 views

An infinite series expansion in terms of the polylogarithm function

We have the complex valued function: $$f(z)=\sum_{n=0}^{\infty}a_{n}\text{Li}_{-n}(z)\;\;\;\;\;\;\;(\left | z\right |<1)$$ We wish to recover the coefficients $a_{n}$. The only thing I though would ...
1
vote
1answer
69 views

Convergent series with coefficient in $\ell^2$.

Let $z$ denote a complex number and $\{\alpha_n\}$ be a sequence in $\ell^2$. Would you help me to prove that series $\sum_{n=0}^{\infty} \alpha_n z^n$ has radius of convergence greater than or equal ...
3
votes
1answer
110 views

Contractive Operator and Realization Theorem

Good morning, I have searched, by using google for a time, a proof of the following theorem : Let $\pmatrix{A&B \\ C&D}\colon H \oplus K \to H\oplus K$ be a contractive operator of a Hilbert ...
3
votes
2answers
284 views

Is a complex space more “advanced” than a “generic” real space?

For instance, does taking the square root of a complex number and its complex conjugate create a metric that "automatically" makes it an inner product space? Is a complex space more complete than a ...
4
votes
0answers
144 views

When functions, analytically continued, carry over certain properties

Let $ \Omega $ be a sufficiently smooth planar region in $ \mathbb{R}^2 $ with spectrum $ \Gamma $ (the set of eigenvalues of the Laplace operator on functions which vanish on the boundary $ \partial ...