0
votes
2answers
44 views

Examples of skew adjoint differential operators

I just need some references which studies examples of skew adjoint differential operators generating unitary strongly continuous groups of operators, and its applications to partial differential ...
4
votes
0answers
98 views

Operators such that $\langle Ax,x \rangle=-\langle x,Ax \rangle$

Let $X$ be a Banach space. We consider the differential equation: $$x'(t)=Ax(t), \ \ \ t\in\mathbb{R}$$ where $A$ is a bounded operator on $X$. If $X$ is a Hilbert space, and $x(t)$ is a solution of ...
0
votes
1answer
45 views

Positive operators in Hilbert spaces

Let $H$ be a Hilbert space. I am just asking if there's some reference which studies operators $A$ with this property: $$\left\langle Ax,x\right\rangle \geq0,$$ for all $x\in H$. And $Ax=0$ whenever ...
1
vote
0answers
25 views

Application of operator theory in ODE and PDE

I am looking for references of applications of operator theory (especially spectral theory) in ODE, PDE and possibly SDE. I have learnt operator theory in the general set up, but only know little ...
3
votes
0answers
113 views

Differentiation of norm in Banach space (explanation of text needed)

Let $Y$ be uniformly smooth Banach space. Consider the convex $C^1$ functional $\Phi:Y \to \mathbb{R}$ defined $$\Phi(y) = \frac{1}{q}\Vert y \Vert^q_{Y}.$$ Its derivative $\varphi:Y \to Y'$ is a ...
1
vote
0answers
21 views

Conjugating an operator with a gauge transformation; how is the kernel affected.

For the differential operator $$ D := i I \frac{d}{dx} + A(x) \colon C^\infty_T([0,\beta]),\mathbb{C}^m) \to C^\infty_T ([0,\beta],\mathbb{C}^m) $$ where $A(x)$ is Hermitian and $C^\infty_T ...
1
vote
0answers
27 views

Smoothness of solutions to Fredholm integral equation

Let $K(x,y)=k(|x-y|)$ where $k$ is continuous on $(0,1]$, and assume function $f\in L^2[0,1]$ satisfies $f(x)=\int_0^1 f(y)K(x,y)dy$. Is $f$ necessarily $C^\infty $ ? under what condition on kernel ...
3
votes
0answers
45 views

operator on separable banach space whose spectrum and point spectrum is prescribed compact set

I am interested in obtaining the following paper: G. K. Kalisch, "On operators with large point spectrum," Scripta Math. 29 No. 3-4, (1973), 371-378. According to Ben Mathes, "Strictly Cyclic ...
4
votes
1answer
67 views

MASAs of C* algebras

While studying the $C^*$-algebraic formulation of the recently solved Kadison-Singer problem, I was wondering about maximal abelian subalgebras: Let $\mathcal{A}$ be a unital C* algebra. There seems ...
0
votes
0answers
42 views

Projection on intersection as strong limit

For orthogonal projections $P,Q$ on some Hilbert space $H$ let $P\wedge Q$ denote the orthogonal projection on $\text{Im }P\cap\text{Im }Q$. Clearly if $P,Q$ commute $P\wedge Q$ is given by $PQ$. I ...
2
votes
1answer
129 views

Learning roadmap for Non-commutative Geometry [closed]

I am interested in learning Non-commutative geometry and K-theory of operator algebras. Please suggest a learning roadmap for this subject. My present knowledge of Measure theory & Functional ...
1
vote
0answers
95 views

Compactness of advection diffusion operator (elliptic operator)

Consider the ADE defined on a compact domain $\Omega \in \mathbb R^n$, with boundary $\partial \Omega$ Assume $u(x)$ is smooth incompressible vector field, with $u(x).\hat n(x)=0$ for $x\in \partial ...
6
votes
3answers
421 views

Books for studying Dirac Operators, Atiyah-Singer Index Theorem, Heat Kernels

I am interested in learning about Dirac operators, Heat Kernels and their role in Atiyah-Singer Index Theorem. From various sources (including this very helpful question), I have come to know of ...
1
vote
2answers
143 views

Operator Theory References and Topics

I wish to do a reading course in Operator Theory. Thus, I am looking for some references in the area. Right now, I have the following two sources available: Unbounded Self-Adjoint Operators ...
0
votes
0answers
57 views

Is an exact operator, unitary equivalent to a banded operator?

Let $H$ be an infinite dimensional separable Hilbert space and $B(H)$ the algebra of bounded operators. Definition : Let $(e_{n})_{n \in \mathbb{N}}$ be an orthonormal basis. $T \in B(H)$ is ...
4
votes
1answer
55 views

Is every irreducible operator unitary equivalent to a banded operator?

This issue continues this question. Let $H$ be an infinite dimensional separable Hilbert space and $B(H)$ the algebra of bounded operators. Definition : Let $(e_{n})_{n \in \mathbb{N}}$ be an ...
5
votes
1answer
168 views

Is every operator unitary equivalent to a banded operator?

Let $H$ be an infinite dimensional separable Hilbert space and $B(H)$ the algebra of bounded operators. Definition : Let $(e_{n})_{n \in \mathbb{N}}$ be an orthonormal basis. $T \in B(H)$ is ...
2
votes
1answer
86 views

Is there an irreducible, noncompact commuting, nonnormal operator, with spectrum strictly continuous?

Let $H$ be an infinite dimensional separable Hilbert space. Definition: The commutant $\mathcal{S}'$ of a subset $\mathcal{S} \subset B(H)$ is $ \{A \in B(H) : AB=BA \ , \ \forall B \in \mathcal{S} ...
4
votes
1answer
128 views

Resource on Infinite Systems of Difference Equations

In my efforts (somewhere on the boundary of discrete mathematics and theoretical computer science), I have come upon the necessity of solving (or at least finding out some of the solution's ...
2
votes
2answers
56 views

Self-adjoint extensions modern paper or book

Do you know some modern and recent paper, lecture notes, or book about self-adjoint extension theory (defect indeces, Von Neumann theory,...)? Classical references can also be helpful but I am ...
10
votes
1answer
341 views

Looking for an easy lightning introduction to Hilbert spaces and Banach spaces

I'm co-organizing a reading seminar on Higson and Roe's Analytic K-homology. Most participants are graduate students and faculty, but there are a number of undergraduates who might like to ...
3
votes
1answer
253 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
215 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 ...
10
votes
1answer
630 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
92 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
228 views

Functional analysis summary

Anyone knows a good summary containing the most important definitions and theorems about functional analysis.
1
vote
1answer
59 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
63 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
51 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
64 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
187 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 ...
5
votes
0answers
342 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
46 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
1answer
138 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
298 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 $$ \vert\vert T \vert\vert= \sup |{\rm tr} (TP)| $$ ...
3
votes
3answers
1k 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
79 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
81 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
304 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
329 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
188 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
225 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
93 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
132 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 ...
7
votes
2answers
1k 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 ...