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Questions tagged [operator-theory]

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

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86 views

Interesting examples of spectral decompositions of BOUNDED operators with both continuous and discrete spectrum

I would like to have a few basic examples of bounded self-adjoint operators $T$ (more generally bounded normal would be fine) on a Hilbert space $(H,\langle,\rangle)$ for which the following criteria ...
2
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0answers
60 views

k-rank numerical range of an operator

Let $T\in\mathscr{B(\mathcal{H})}$ where $\mathcal{H}$ is an infinite dimensional seperable Hilbert Space and $k\in\mathbb{N}\cup\{\infty\}$. Now we define k-rank numerical range of $T$ denoted by $\...
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0answers
46 views

Convolution with an analytic semigroup

Let $e^{At}$ denote an analytic semigroup on Hilbert space $X$ generate by $A:D(A)\to X$. Also, let $f\in L^1(0,\tau;X)$. I want to show that the convolution $$ g(t)=\int_0^t e^{A(t-s)}f(s)ds$$ ...
1
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1answer
91 views

When do supremum and expectation commute?

This is an alternative form of the question in When do maximum and expectation commute? When we looking for conditions on $G(t,x(t))$ such that $$ \sup\limits_{t\in [0,N]}E[G(t,X(t))]=E[\sup\limits_{...
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0answers
90 views

Is this basis a Schauder basis?

Let $G$ be a torsion free group. Let $\alpha$ be an element in $\mathbb CG$, the group algebra of $G$, with $\|\alpha\|_1=1$ and assume that $\{1,\alpha,\alpha^2,\dotsc\}$ is linearly independent, $(...
1
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1answer
81 views

Does this element belong to all powers of the augmentation ideal of the group algebra.

Let $G$ be a torsion free group, and let $\alpha$ and $\beta$ are elements in the augmentation ideal, $I$, of $\mathbb CG$, the group algebra of $G$. Assume that there exists complex numbers $a$ and $...
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0answers
68 views

Are these kinds of “crossed product” studied?

Let $M$ be a von Neumann algebra acting in a Hilbert space $H$, and let $\rho$ be a representation of a group $G$ on a Hilbert space $K$. Define $M\rtimes_\rho G$ to be a von Neumann algebra acting in ...
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0answers
53 views

About crossed product of the group von Neumann algebra

Let $G$ be a group with a normal subgroup $N$. If $K$ is a field, then it is well known in ring theory that $K[G]\cong K[N]*(G/N)$ ($*$ stands for the crossed product). Do we have such an isomorphism ...
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1answer
67 views

Convergence of self-adjoint elements in $\sigma$-weak topology

In a von Neumann algebra, if $A_{\alpha}$ converges to $0$ in the $\sigma$-weak topology, do the positive parts $(A_{\alpha})_{+}$ necessarily converge to $0$ in the $\sigma$-weak topology?
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45 views

Theta Summable operator with bounded trace

Let $D$ be an unboudned self-adjoint operator on the Hilbert space $H$. We assume that all spectrum of $D$ are eigenvalues and $D$ is theta-summable, i.e. $e^{-tD^2}$ is of trace class for all $t>...
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2answers
148 views

Compact images of nowhere dense closed convex sets in a Hilbert space

Let $B=-B$ be a nowhere dense bounded closed convex set in the Hilbert space $\ell_2$ such that the linear hull of $B$ is dense in $\ell_2$. Question. Is there a non-compact linear bounded operator ...
3
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0answers
80 views

eta invariant and spectral flow

We know that for a family of first-order self adjoint elliptic (Fredholm) operator $A_t$, for $t\in [0,1]$ we have the formula $$\eta(A_1)-\eta(A_0)=spfl(A_t)_{t\in[0,1]}+\int^1_0 \omega(s)ds,$$ where ...
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0answers
70 views

Bounded and sectorial operators

Is there any assumption for a bounded operator to be sectorial ? Is there any characterization of such operators ? Here, the definition of sectorial operators follows the book of Markus Haase: ...
1
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1answer
136 views

When is $\inf_{n\geq0}x^n\neq0$?

Let $M$ be a von Neumann algebra acting on a Hilbert space $H$. Let $x$ be a positive element of $M$ with $\|x\|=1$. So, $(x^n)_{n\in\mathbb N}$ is a decreasing sequence of positive elements and $y:=\...
2
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1answer
75 views

Generation of strictly contraction Semigroups

Let $T(t)$ be a $C_0$-semigroup on Banach space $X$, and $A$ its generator. By Lumer-Philipps theorem we know that if $A$ is densely defined and m-dessipative operator then it generates a $C_0$-...
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1answer
54 views

On the eigenvalue of the expectation value of a random matrix in quadratic form

When we handle with some dynamic input-output mappings, there occurs a question as follows: Let $M$ be a random matrix, of which each element contains random terms. Consider the two expectation ...
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0answers
27 views

A variation on Sylvester equation

Let $X$ be a finite measure space and $D,M$ be bounded linear operators on a $(L^1(X;\mathbb C))^2$. $D$ is a diagonal operator matrix whose entries are multiplication operators by the invertible ...
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0answers
29 views

Relative compactness of differential operator

Let $\Omega$ be $\mathbb R^n$ or a complete (unbounded) open manifold, and $f$ be a smooth function on $\Omega$. We consider a self-adjoint 2nd. elliptic operator $H$ on $L^2$ space(to simplify the ...
14
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2answers
447 views

Multiple of identity plus compact

Is there an example of a bounded operator $T\in\mathcal{B}(H)$, where $H$ is a separable complex Hilbert space, such that no restriction to an infinite dimensional closed subspace is multiple of ...
2
votes
1answer
99 views

Regarding approximation by invertible operators [closed]

This post here states that if $E$ is an infinite dimensional space and if $T$ is an injective, bounded,non surjective opertor with closed range in $E$, then $T$ cannot be approximated in operator norm ...
2
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1answer
68 views

Dirac operator on manifold with periodic end

Let $\tilde W$ be a spin closed oriented manifold, $Y$ is a codimension $1$ closed oriented submanifold of $\tilde W$, and denote the $W$ the cobordism from $Y$ to itself obtained from cutting $\...
2
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1answer
60 views

Regarding norm attaining functions

Let $X$ and $Y$ be Banach spaces.Let $L(X,Y)$ denote the space of all bounded linear map from $X$ to $Y$. $T:X\longrightarrow Y$ is said to be norm attaining if there exists a $x\in S_X$(the closed ...
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0answers
104 views

Renyi's theorem on mixing

I have been trying to understand the proof of Renyi's characterization of (strongly) mixing transformations: A measure preserving transformation $T \text{ is strongly mixing iff for every measurable }...
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0answers
61 views

Support size of a zero divisor

Let $G$ and $\mathbb C[G]$ be a torsion free group and its group algebra. Is there a function $f:\mathbb N\rightarrow\mathbb R$, with $\lim_nf(n)=\infty$ such that if $0\neq\alpha,\beta\in\mathbb C[G]$...
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175 views

Constructing a noncommutative algebra from a commutative algebra

I was told at a conference that one way to construct a noncommutative algebra from a commutative one is to "replace the product of finite spaces (which on the level of continuous functions corresponds ...
1
vote
1answer
89 views

Optimal estimate in trace norm

Let $x,y$ be vectors of some Hilbert space of unit length. Then we can consider the projection $P_x:=\langle \bullet, x \rangle x$ and similarly $P_y.$ Assume then that we know that $\left\lVert x-...
2
votes
0answers
102 views

An example of non trivial projections in a group von Neumann algebra

Let $G$ and $\text{vN}(G)$ be a torsion free group and its group von Neumann algebra. Is there a characterization of non trivial projections in $\text{vN}(G)$? If not, is a certain class of them ...
3
votes
1answer
142 views

An analytical zero divisor

Let $G$, $\mathbb C[G]$ and $\text{vN}(G)$ be a torsion free group, it's group ring and group von Neumann algebra, resp.. Let $0\neq\alpha\in\mathbb C[G]$ and $0\neq p\neq1$ is a projection in the ...
4
votes
0answers
63 views

Superposition operator from Sobolev space to Lebesgue space

Let $\Omega$ be a bounded, connected set in $\mathbb{R}^2$ with smooth boundary. I am wondering under what conditions on the real function $f(x):\mathbb{R}\to \mathbb{R}$ the superposition operator $F(...
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0answers
121 views

Compact embedding result

Let $\tau$ and $\ell$ be positive numbers. We know that the space $H^2(0,\ell)\cap H^1_0(0,\ell)$ is compactly embedded into $L^6(0,\ell)$. Now, is the space $L^2(0,\tau;H^2(0,\ell)\cap H^1_0(0,\ell))$...
2
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1answer
81 views

Fredholmness of formal selfadjoint operator $AA^*$ and Fredholmenss of $A$

Let $X$ and $Y$ be Hilbert spaces with respective inner products $\langle , \rangle_{X,Y}$. Let $A:X \rightarrow Y$ be a bounded linear operator. Assume there is a non-degenerate sesquilinear product $...
1
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1answer
55 views

Disreteness of spectra of Dirichlet laplacians

I have fundamental questions on Dirichlet Laplacians. Let $D \subset \mathbb{R}^d$ be an open subset and $\mathcal{L}$ be the (non positive) Dirichlet laplacian on $D$. We denote by $T_t=e^{t\...
3
votes
1answer
106 views

Schrödinger operator with Coulomb potential

The free Laplacian $-\Delta$ has absolutely continuous spectrum $[0,\infty).$ The Coulomb Hamiltonian $H=-\Delta-\frac{1}{\vert x\vert}$ on $L^2(\mathbb R^3)$ has absolutely continuous spectrum $[0,\...
4
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1answer
108 views

Energy in doubling measure metric spaces

Let $(X,\mu, d)$ be a metric measure space where $\mu$ is a doubling measure. For a relatively compact set $U\in X$ consider the following quantity $$I(U,\mu,d)=\int_U \int_U \log^2(d(x,y)) d\mu(x) d\...
10
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1answer
255 views

Criterion for a Banach algebra to be finite dimensional

Let $A$ be a Banach algebra (say, complex and unital) and suppose that every (closed) commutative subalgebra of $A$ is finite dimensional. Question. Does it follow that $A$ is finite dimensional? ...
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2answers
294 views

Does point-wise weak convergence give weak convergence in $L^2(I;X)$?

Let $X$ be a separable reflexive Banach space, $F$ be a locally Lipschitz nonlinear operator on $X$ that is weakly continuous on $X$, and $u_n$ are $u_n$ weakly converges to $u$ on $L^2(0,T;X)$. Now, ...
2
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0answers
57 views

Common eigenvector of commuting unbounded Operators

Let be T,S two self adjoint linear Operator on a Hilbert Space $\mathcal{H}$ with pure point spectrum. Then T,S commute if and only if they have a complete set of common eigenvectors. This is the ...
3
votes
1answer
155 views

Is the kernel of a Fredholm operator stable under perturbation?

This is a follow-up of this question. In a nutshell: Does the kernel of a bounded operator change "nicely" with the operator? Let $(X,\| \cdot \|)$ be an infinite-dimensional normed vector space. ...
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0answers
77 views

Conditional Expectation for von Neumann algebra

Let $M$ be a von Neumann algebra and $T: M\rightarrow \mathbb{C}$ be a finite normal faithful tracial map, s.t., if $\phi : M \rightarrow A \cap A^{*}$ is a conditional expectation, (A being a weak ...
0
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1answer
182 views

Elementary quantum scattering problem on the line.

Let us consider the quantum scattering problem on the line with the Hamiltonian $$H=-\frac{d^2}{dx^2}+ V(x),$$ where $V(x)=1$ when $x\in (0,a)$, and $V(x)=0$ otherwise. It is easy to see that $H$ ...
4
votes
1answer
184 views

Can this self-adjoint operator have an infinite-dimensional compression with compact inverse?

The following might be quite straightforward, but I very rarely work in detail with unbounded operators, so I thought it would be worth seeing quickly if I have overlooked an example that is obvious ...
4
votes
1answer
104 views

Closure of tensor product /tensor product semigroup

In this reference the following claim is made in Remark 2 Let $A,B$ be closable operators on Banach spaces $X,Y$, then $A \otimes 1$ and $1 \otimes B$ are closable operators on the Banach space $X \...
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0answers
33 views

Invariance of simple functions

Let $(A(s),D(A(s)))_{s\in\mathbb{R}}$ be a family of unbounded operators on a Banach space $X$ and $g:\mathbb{R}\rightarrow X$ be a simple function, i.e., \begin{align*} g=\sum_{i=1}^n{x_i\textbf{1}_{...
8
votes
2answers
295 views

Hölder continuity for operators

Let $x,y$ be positive real numbers then $$|\sqrt{x}-\sqrt{y}|=\dfrac{|x-y|}{\sqrt{x}+\sqrt{y}}=\sqrt{|x-y|}\cdot \dfrac{\sqrt{|x-y|}}{\sqrt{x}+\sqrt{y}}\leq 1\cdot |x-y|^{\frac{1}{2}}$$ we obtain $1/...
3
votes
0answers
39 views

Is singular Cauchy operator bounded in Morrey spaces?

The singular Cauchy operator is defined by $$S_\Gamma :f \to \int_\Gamma \frac{f(\xi)}{\xi-z} d\xi , z\in \Gamma.$$ Is this operator bounded in Morrey spaces and weighted Morrey spaces? i.e. is there ...
11
votes
1answer
324 views

One question about the $\eta$ invariant

This question is from the paper, The Analysis of Elliptic Families II. Dirac Operators, Eta Invariants, and the Holonomy Theorem, Commun. Math. Phys. 107, 103-163 (1986) --- Proposition 2.8. Suppose ...
1
vote
1answer
105 views

The tensor product of two bounded operators

Let $E$, $F$ be two complex Hilbert spaces and $\mathcal{L}(E)$ (resp. $\mathcal{L}(F)$) be the algebra of all bounded linear operators on $E$ (resp. $F$). The algebraic tensor product of $E$ and $F$ ...
2
votes
0answers
181 views

Absence of fixed points

Let $f$ be an arbitrary function in $L^2(0,\infty)$ and consider the function $$(g_f)(y) = \frac{1}{y-x_0} \int_{0}^{\infty} f(x) \frac{xy}{(x^2+y^2+1)} \ dx$$ where $x_0$ is an arbitrary but fixed ...
3
votes
0answers
96 views

Compactness of operators of Markov processes

I have a question about operators of Markov processes. Let $(E,\mu)$ be a locally compact separable metric measure space. Let $X=(X_t,P_x)$ be a $\mu$-symmetric Markov process on $E$. For any bounded ...
3
votes
1answer
248 views

Function square-integrable

Let $f$ be an arbitrary function in $L^2(0,\infty)$ and consider the function $$(g_f)(y) = \frac{1}{y-x_0} \int_{0}^{\infty} f(x) \left(\frac{xy}{(x^2+y^2+1)}\right)^2 \ dx$$ where $x_0$ is an ...