Tagged Questions

3
votes
1answer
230 views

Sets $f_n\in A_f$ where $f_{n+1}=f_n \circ S \circ f^{\circ (-1)}_n$ and operator $\alpha(f_n)=f_{n+1}$

Let's start with a function on the Reals (in this case for $x=0$ is not defined): for example $f(x)=b/x$, $x \in \mathbb R$ I define: $$f_0:=f$$ $$f_{n+1}:=f_n \circ S \circ f^{\circ ...
2
votes
1answer
90 views

Gelfand's Formula. $r(T)=\lim_{n \to\infty}\sqrt[n]{\|T^{n}\|}$

Can you indicate me a material where I cand find the proof of Gelfand's Formula. I heard that there is a proof with polynomials. Gelfand's Formula : If $T \in B(X)$ then: $$r(T)=\lim_{n ...
8
votes
1answer
173 views

Importance of Toeplitz operators?

I am reading Arveson's A Short Course on Spectral Theory, in which the author states that Toeplitz operators are very important without giving references on their applications. After some searching, I ...
3
votes
1answer
71 views

Does $|\langle x,Ax\rangle |=\langle x,|A|x\rangle$ for bounded operators on Hilbert space?

If $A$ is a bounded linear operator on a Hilbert space $H$ is it true that $|\langle x,Ax\rangle |=\langle x,|A|x\rangle$ for all $x\in H$? If not, can we at least establish inequality in one ...
3
votes
1answer
121 views

Functional analysis summary

Anyone knows a good summary containing the most important definitions and theorems about functional analysis.
1
vote
1answer
36 views

Reference about Fredholm determinants

I am searching for a reference book on Fredholm determinants. I am mainly interested in applications to probability theory, where cumulative distribution functions of limit laws are expressed in terms ...
1
vote
0answers
49 views

Containment of an element to an operator system

This question will probably appeal to people in operator systems theory as it is very much related. However, I'm interested in down-to-earth concrete systems with finite dimensional Hilbert space ...
0
votes
1answer
44 views

Does every operator have a matrix?

Maybe this question is basic however I'm not familiar with operator theory. Does every operator have a matrix? I also would like to see some proof of this fact (if it's elementary) or at least get ...
1
vote
0answers
56 views

Positive maps on $\mathcal{B}(\mathcal{H})$ to itself

Let us consider the set of positive maps $\phi:\mathcal{B}(\mathcal{H})\rightarrow \mathcal{B}(\mathcal{H})$ ($\mathcal{H}$ is Hilbert space). Can we characterize all the maps which satisfies the ...
2
votes
1answer
110 views

Extension of Choi's theorem on extreme completely positive maps

In this paper Man-Duen Choi gave a criteria for a completely positive map to be extreme. For convenience I am writing it below. Let $\phi:\mathcal{M}_n\rightarrow\mathcal{M}_m$. Then $\phi$ is ...
6
votes
0answers
191 views

Min Max Principle and Rayleigh-Ritz-Method for eigenvalues of unbounded operators?

Finding eigenvalues of matrices using the Rayleigh-Ritz quotient is well-known, c.f. http://en.wikipedia.org/wiki/Min-max_theorem Does the following generalization of that fact also hold? Theorem: ...
1
vote
0answers
45 views

Similarity orbit of compact operators

I am considering a problem connecting the spectra of compact operators to larger class of operators. Since spectra are invariant under similarity, I wonder whether there is a good reference on ...
1
vote
0answers
91 views

norm of operator in Hilbert space and complex conjugate Banach space

Let $E$ and $F$ be complex Banach spaces. We denote by $\overline{E}$ the compex conjugate of $E$, that is, the vector space $E$ with the same norm but with the conjugate multiplication by a complex ...
3
votes
2answers
195 views

norm of a normal operator using projections

Let $H$ be a Hilbert space and $T$ a normal operator on $H$. In the sequel, ${\rm tr}$ denotes the trace for trace class operators. Do we have $$ \vert\vert T \vert\vert= \sup |{\rm tr} (TP)| $$ ...
1
vote
3answers
351 views

An introductory textbook on functional analysis and operator theory

I would like to ask for some recommendation of introductory texts on functional analysis. I am not a professional mathematician and I am totally new to the subject. However, I found out that some ...
3
votes
2answers
68 views

questions on convolutors on $L^p(G)$

Let $G$ be a locally compact group. Suppose $1<p<\infty$. We denote by $CV_p(G)$ the space of operators $T$ on $L^p(G)$ such that $T(f*g)=(Tf)*g$. 1) In the book "Amenable locally compact ...
1
vote
0answers
64 views

Where to find Kelly's thesis on Weighted Shifts on Hilbert Space?

I am reading about weighted shifts on a hilbert space. So many of the books/ papers list R.L. Kelly's paper Weighted Shifts on Hilbert Space as a reference that I really want to have a look at this ...
4
votes
0answers
181 views

Eigenprojection as Contour Integral over Resolvent

Let $H$ be a Hilbert space and let $A \in L(H)$ be a bounded linear operator. Assume that $\lambda$ is an eigenvalue of $A$ and assume further that $C_\lambda$ is a simple closed curve in the complex ...
13
votes
1answer
277 views

How does $\sigma(T)$ change with respect to $T$?

Consider $\sigma$ as a mapping which maps $T\in\mathcal{L}(X)$ to $\sigma(T)$, the spectrum of $T$, a compact set in the complex plane. I wonder whether there is some result concerning how ...
7
votes
2answers
138 views

Spectra of restrictions of bounded operators

Suppose $T$ is a bounded operator on a Banach Space $X$ and $Y$ is a non-trivial closed invariant subspace for $T$. It is fairly easy to show that for the point spectrum one has ...
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 ...
1
vote
0answers
78 views

Introductory book on Distribution theory [duplicate]

Possible Duplicate: Distribution theory book Is there a good alternative to Friedlander Introduction to the Theory of Distributions ? Many thanks !
3
votes
0answers
108 views

Fixed point: general case

This is the second part of the question Fixed point: linear operators. Here I would like to ask you about the general case. A lot of concepts can be described or even defined as a solution of a ...
5
votes
1answer
809 views

What is operator calculus?

I watched the excellent interview with Richard Feynman: http://www.youtube.com/watch?v=PsgBtOVzHKI In the interview Feynman mention that he at young age re-invented operator calculus. I have searched ...