0
votes
1answer
22 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
65 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
36 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
0answers
39 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
48 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
262 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 ...
7
votes
0answers
314 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 ...
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
102 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
254 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
143 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 ...