Spectrum, resolvent, numerical range, functional calculus, operator semigroups. Special classes of operators: compact, Fredholm, dissipative, differential, integral, pseudodifferential, etc.

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36
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1k views

Set-theoretic reformulation of the invariant subspace problem

The invariant subspace problem (ISP) for Hilbert spaces asks whether every bounded linear operator $A$ on $l^2$ (with complex scalars) must have a closed invariant subspace other than $\{0\}$ and ...
14
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0answers
419 views

Unital $C^{*}$ algebras which all elements have path connected spectrum

A unital $C^{*}$ algebra is called "Path connected" if the spectrum of all its elements is a path connected subset of $\mathbb{C}$. Is the tensor product of two path connected algebra, a path ...
10
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399 views

Functional calculus of unitary matrices and commutator norms: reference request

Suppose we have a normal matrix $A$ and a general matrix $B$. For a continuous function $f$ on the disk we can find upper-bounds on $\Vert[f(A),B]\Vert$ in terms of $\Vert[A,B]\Vert$. The more we know ...
8
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70 views

Holomorphic natural bundles and operators

I am wondering up to what extent the classical theory of (smooth) natural bundles and natural operations extends to the holomorphic setting. After a quick thought, I've gone through the standard ...
6
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127 views

Is there a quotient of $c_0$ without the approximation property?

The famous example of Enflo of a Banach space without the approximation property is actually a subspace of $c_0$. Is there a quotient of $c_0$ without the approximation property? This would follow if ...
6
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184 views

Operator arithmetic-harmonic mean inequality with operator-valued weights

Let $\Lambda_1,\dots,\Lambda_n$ be strictly positive definite operators in the Euclidean space $\mathbb{R}^d$. By an operator arithmetic-harmonic mean inequality with weights $\Lambda_i$ I mean the ...
6
votes
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103 views

How to take this Grassmann integral?

I'm trying to reconstruct and understand what is explained in a paragraph of this paper. I am trying to check if the method they describe actually gives us the Laughlin state. The integral I'm facing ...
6
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239 views

What is the intersection of the closures of left invertible operators and right invertible operators?

From Douglas Zare's answer (see Does $X$ embed in $Y$, and $Y$ embed in $X$, always imply that $X$ isomorphic onto $Y$?), one know that $$ \overline{G_{l}(X,Y)} \bigcap \overline{G_{r}(X,Y) } = ...
5
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184 views

Eigenvalues of a certain product of matrices with special structure

Let $d$ and $c$ be positive integers and $q = dc$. Let $G$ be a $q$-by-$q$ positive semi-definite real matrix with eigenvalues all $\le 1$, and define the $q$-by-$2q$ matrix $A = ...
5
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99 views

Fock Space Proof of $(g(x)\phi^4)_2$ Mass Gap?

Is there a proof that does not depend on Euclidean methods? Is this a proof? : $V(g)$ can be written as $P+R$ where $P$ is non-negative and $R$ is $N$-bounded (and hence $(H_0+\lambda P)$-bounded). ...
5
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119 views

When is an inner derivation a Fredholm operator?

Let $\mathcal{B}(H)$ denote the algebra of bounded operators on a Hilbert space $H$. I'm interested in inner derivations acting on the Schatten ideals $L^p\subseteq\mathcal{B}(H)$ (defined by ...
4
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65 views

Some questions on the nodal geometry of Dirac operators

Let me begin by quoting a well-known result of Christian Baer (see here). The result goes as follows: Theorem (Baer): Consider a connected $n$-dimensional Riemannian manifold with Dirac bundle $S$ ...
4
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95 views

Smooth perturbation of a positive self-adjoint operator with compact resolvent

Consider a one-parameter family $A_t$ of unbounded positive self-adjoint operators with discrete spectrum (for example, one can consider a one-parameter family of Laplacians on a compact Riemannian ...
4
votes
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86 views

Domain of $A^{1/2}$ on $L^2(\mathbb{T}^2)$

Let $A$ be a densely defined unbounded self-adjoint operator defined on $L^2(\mathbb{T}^2)$, where $\mathbb{T}^2$ stands for the 2-torus. It is known that $A$ is positive, that is, $\langle Au, ...
4
votes
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109 views

Infinitesimal Generator of Billiard Flow

The Billiard flow $S_t$ on a Riemannian manifold with boundary (with corners) is the group of operators defined on continuous functions on the Co-sphere bundle as follows: To determine $S_t u(\xi)$, ...
4
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0answers
133 views

A question on extension of $Z^{*}$ algebras

A $Z^{*}$ algebra is a $C^{*}$ algebra which all elements are(two sided or equivalently one sided) zero divisor. Are there two $Z^{*}$ algebras $A,B$ such that for every short exact sequence ...
4
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141 views

The representation-theoretic nature of an operator resolvent

Consider parameter $s$ in definition of $R(s,A)=(s I - A)^{-1}$ where $A$ is a linear operator in a vector space $X$. When $X$ is over $\mathbb{C}$, then $s$ is thought to be a complex number. Now ...
4
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317 views

Denseness of finite rank operators in $\mathcal{B}(X,Y)$

Let $X$ and $Y$ be Banach spaces and let $\mathcal{B}(X,Y)$ be the space of bounded linear operators from $X$ to $Y$. As noted in the answers to a question on ...
4
votes
0answers
187 views

Adjoint of sum of two operators. Kato-Rellich

Let $A$ be self-adjoint and $B$ be symmetric with $A$-bound less than $1$. By Kato-Rellich, I know that $(A+B)^*=A+B$. Could I also get something like $(A+iB)^*=A-iB$ or is there a counterexample to ...
4
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0answers
119 views

Noncommutative Poincaré inequalities

This is a question on how (or if) people in the community think about the Poincaré inequality in noncommutative geometry. In geometry, the Poincaré inequality (when it exists) gives a bound on a ...
4
votes
0answers
70 views

Strictly convex renormings making power bounded operators into contractions

Let $X$ be a Banach space and let $T$ be a power bounded linear operator on $X$ (i.e. $\sup_{n\ge0}\|T^n\|_{op}<\infty$). We can of course define an equivalent norm $\|\cdot\|'$ on $X$ so that ...
4
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209 views

Essential unitary equivalence

Let us agree that heuristic meaning of the word "essential" is: up to compact operator. There is clear notion of unitary equivalent operators. What is the proper notion of two operators being ...
4
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0answers
450 views

About generator and isomorphism problems for free groups operator algebras

Let $H$ be an infinite dimensional separable Hilbert space. The $C^{*}$-algebras and von Neumann here unital and subalgebras of $B(H)$. Definition : Let $\mathcal{A}$ be $C^{*}$-algebra (resp. a ...
4
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477 views

Do the banded operators check the invariant subspace problem?

Let $H$ be an infinite dimensional separable Hilbert space and $B(H)$ the algebra of bounded operators. Invariant subspace problem: Let $T \in B(H)$. Is there a non-trivial closed $T$-invariant ...
4
votes
0answers
113 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 ...
4
votes
0answers
594 views

The spectrum of a Markov Operator and Invariant Measures

Suppose I have a discrete-time Markov Chain (in an infinite dimensional state space $\Omega$) with Markov operator $P$, a linear operator on the space of bounded measurable functions on $\Omega$. (Or ...
3
votes
0answers
55 views

Spectral theory of Bochner integral operators

Consider the following (somewhat simplified) situation. Let $\mathcal{H}$ be separable Hilbert space and $\mathcal{B}(\mathcal{H})$ the Banach algebra of bounded linear operators acting on ...
3
votes
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223 views

Graded structures for simple $C^{*}$ algebras without nontrivial idempotent

Edit(A confession): I just realized that the question is trivial: Since one can easily prove that the convex hull of the spectrum of every nontrivial homogeneous element of a $\mathbb{Z}_{n}$-graded ...
3
votes
0answers
38 views

Reference request: derivative of trace of heat operators with respect to a parameter

I hope that the following question is appropriate to ask here as it is not exactly research or original Mathematics but rather an enquiry for a reference or "standard method of proof": Suppose we are ...
3
votes
0answers
111 views

Stability of a linear system and spectrum of the product of two matrices

Let us consider an invertible matrix $\mathbf{A}\in GL_d(\mathbb{R})$ such that all its diagonal entries $\mathbf{A}_{ii}=-1 \; \forall \, i$. My question is the following: Does it always exists a ...
3
votes
0answers
208 views

The Dunkl intertwining operator $V_k$ on $C(\mathbb{R}^d)$

The Dunkl intertwinig operator $V_k$ on $C(\mathbb{R}^d)$ is defined by: $$V_k f(x)=\int_{\mathbb{R}^d}f(y)d\mu_x(y),$$ where $d\mu_x$ is a probability measure on $\mathbb{R}^d$ with support in the ...
2
votes
0answers
50 views

The motivation of Weyl-Titchmarsh function

Given a second linear differential operator, $(Hf)(x)=-\frac{d^2}{dx^2}(x)+V(x)f(x)$, where $V$ is a bounded and real valued function, $f$ lies in $L^2(\mathbb{R})$. For an $z$ with $Im(z)\neq ...
2
votes
0answers
141 views

Discrete p-Laplacian

One of the definitions of the discrete (weighted) $p$-Laplacian is the following: $$\Delta_{p,w}u(x):=\sum_y |u(y)-u(x)|^{p-2}(u(y)-u(x))w(x,y).$$ Consider the one dimensional case. Then the free ...
2
votes
0answers
127 views

Discrete versus Continuous Hilbert Transform

Let me define the Fourier transform of a function $u$, say in the Schwartz space $\mathscr S(\mathbb R)$ as $ \hat u(\xi)=\int_{\mathbb R} e^{-2iπ x\cdot \xi} u(x) dx. $ The Hilbert transform ...
2
votes
0answers
83 views

Semigroups on Banach Lattice

Let $\{Z(t)\}_{t\geq 0}$ be a semigroup of positive operators on a Banach lattice $X$. I want to show that $$\int_0^1[Z(1-s)+Z(1+s)]fds-f\in X_+,\quad f\in X_+$$ Where $X_+$ denotes the positive ...
2
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0answers
81 views

Necessity of coercivity assumption in Minty's theorem

Minty's Theorem states that a bounded, Continuous, monotone and coercive function on a Hilbert space is surjective. A function $f:H\to H$ is called coercive if $$ \lim_{\|x\|\to \infty} \frac{\langle ...
2
votes
0answers
269 views

Predual of a von Neumann algebra in terms of trace class operators

For a von Neumann algebra $\mathcal{A} \subseteq \mathcal{B(H)}$ where $\mathcal{B(H)}$ is the space of all bounded linear operators on the Hilbert space $\mathcal{H}$, there is a Banach space $ ...
2
votes
0answers
87 views

Constant in Maximal sobolev regularity

We know the following evolution equation \begin{equation} \left\{ \begin{array}{llc} v_t=A v+f,\\ v(0)=0. \end{array} \right. \end{equation} $A$ generates a bounded analytic semigroup on a Banach ...
2
votes
0answers
53 views

Is the Szego projection on a codim-$k$ CR manifold an integral operator?

The Szego projection on a CR manifold $M$ is defined to be the orthogonal projection from $L^2(M)$ to the closed subspace $H^2(M),$ where $$H^2(M) = \{f \in L^2(M)\ |\ \bar{\partial}_{b}f = 0\ ...
2
votes
0answers
74 views

series representation for *un*bounded perturbations of semigroup generators

Let $A$ generate an analytic $C_0$-semigroup on a Banach space $X$ and $B$ be a relatively compact perturbation, i.e., $B$ is compact as an operator from $D(A)$ (with the graph norm) to $X$. Then ...
2
votes
0answers
243 views

Versions of the spectral theorem

Since any $C^*$-algebra can be represented as an algebra of bounded operators $\mathcal{B(H)}$ on a Hilbert space $\mathcal{H}$, the spectral theorem applies to all $C^*$-algebras: ($*$) ...
2
votes
0answers
182 views

Invariant Measures of Markov Chains under Perturbations

This is a more specific version of a question I asked before without much luck. I believe this should be standard perturbation theory, but looking at Kato's book has not helped. Any references would ...
2
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0answers
99 views

non-closed weak graph limit of symmetric operators

Hi Everyone, I was recently reading Reed & Simon's functional analysis textbook (the first volume), and it mentions casually on page 294 that weak graph limits of a sequence of symmetric ...
2
votes
0answers
194 views

A specific projection and compactness on the Bargmann-Fock space

Let $F_2$ be the Bargmann Fock space defined as the space of entire functions $f$ on $\mathbb{C}$ such that \begin{align*} \int_{\mathbb{C}} |f(z)|^2 e^{- |z|^2} dA(z) \end{align*} ($dA$ is just ...
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0answers
69 views

Heat semigroup ultracontractive?

Let $g(x,t)= \frac{1}{(4 \pi t)^{\frac{n}{2}}}e^{\frac{-|x|^2}{4t}}$ be the heat kernel on $\mathbb{R}^{n}.$ Is the standard definition now to say that this heat-semigroup $T(t)(f):=g *f(.,t)$ is ...
1
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0answers
71 views

One-parameter group of unitary operators and Core

Question : For what condition on $V$ (we can take it smooth, bounded, whatever necessary), the one-parameter unitary group $U(t)$ associated to the seladjoint operator $A=-\Delta+V$ on $\mathbb{R}^n$ ...
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0answers
40 views

Factorization of the Fokker-Planck semigroup

I have posted this question on stackexchange over four month ago, but didn't get an answer: "In the classical theory of Markov processes, the Fokker-Planck semigroup $\{T_t:t\ge 0\}$ can be factored ...
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0answers
40 views

How to define Biharmonic operator for second order sobolev spaces

I am studying an article Link of Article. There author assumes that $\Omega \subset \mathbb{R}^N$, $ N>4 $ . Some where in the paper we have $$ \Delta^2 (\cdot) - \frac{\lambda}{|x|^4} (\cdot) : ...
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102 views

Homomorphic Commutator? Equation

So I was considering the following functional equation: Given $H :\Bbb{C}^2 \rightarrow \Bbb{C} $ find $\theta: \Bbb{C}^2 \rightarrow \Bbb{C}$ such that $$ \theta(H(a,b), H(c,d)) = H(\theta(a,c), ...
1
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0answers
86 views

Generating a series representation for the inverse of the operator $f(f)$

I am considering the following problem: Suppose you are given a function $u: C \rightarrow C$, find a function $g$ such that $g(g) = u$ (Let's assume that such a function exists). And by "find", I ...