Tagged Questions

1
vote
1answer
44 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
29 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 ...
5
votes
0answers
55 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
122 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
58 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 ...
4
votes
0answers
85 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
149 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
48 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
218 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
70 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
205 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
148 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
148 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 ...