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Existence of orthonormal basis for $L^2(G)$ in $C_c(G)$

Remark: I cross-posted this question on MSE and added a bounty to it. Suppose that $G$ is a locally compact (Hausdorff) group endowed with the Haar measure. It is well-known that the compactly ...
Calculix's user avatar
  • 321
11 votes
0 answers
389 views

Von Neumann Inequality in Banach spaces

It is known that the only Banach space that satisfies the von-Neumann inequality is the Hilbert space: Theorem (see e.g. Pisier, "Similarity Problems and Completely Bounded Maps", p 27) For a Banach ...
erz's user avatar
  • 5,529
11 votes
0 answers
529 views

Contraction semigroup on Hilbert space

I'd like to know whether a certain unbounded operator on a Hilbert space is the generator of a strongly continuous contraction semigroup. (Such operators are known as maximally dissipative operators.) ...
André Henriques's user avatar
10 votes
0 answers
225 views

Can the trace be computed in any Schauder basis?

I'm cross-posting this question from Math.SE, as it didn't get much attention there. Let $H$ be a separable Hilbert space and $T \in L(H)$ a trace-class operator. It is well known that the trace of $T$...
WillG's user avatar
  • 233
8 votes
0 answers
251 views

Struggling with a proof in the preprint 'Hermitian geometry on resolvent set'

I have been struggling for awhile with a particular argument in the paper below. I posted the question first on MathSE, but I got no answers. I understand however that MO might be an overreach for ...
WeakMath's user avatar
4 votes
0 answers
160 views

Solution without using any k-theory tools

Let $A$ be the UHF-algebra of type $2^{\infty}$. Suppose that $p$ and $q$ are two projections in $A$ and $\tau(p) = \tau(q)$, where $\tau$ is the unique normalized trace. Then there is a partail ...
Peg Leg Jonathan's user avatar
4 votes
0 answers
2k views

Eigenvalues and spectrum of the adjoint

In a finite-dimensional Hilbert space, the eigenvalues of the adjoint $A^*$ of an operator $A$ are the complex conjugates of the eigenvalues of $A$. But in infinite dimensions this need no longer be ...
Arnold Neumaier's user avatar
3 votes
0 answers
95 views

Commutator of $A\otimes I$ and $I \otimes B$ vanishes?

Consider two Hilbert spaces $H_1$ and $H_2$, and $A$, $B$ unbounded operators on $H_1$, $H_2$ respectively. $(A \otimes I)$ is classically defined as the closure of the operator defined on the set of ...
Hugo's user avatar
  • 31
3 votes
0 answers
198 views

On a paper of von Neumann

Let $H$ be a Hilbert space and $T: H \to H$ be a contraction. In Eine Spektraltheorie für allgemeine Operatoren eines unitären Raumes, von Neumann proved the inequality $$ \lVert p(T)\rVert \leq \sup \...
HaSa's user avatar
  • 31
3 votes
0 answers
122 views

Domain of operator which is used in operator monotone function

We are studying the paper Rupert L. Frank, Leander Geisinger, Refined semiclassical asymptotics for fractional powers of the Laplace operator, Journal für die reine und angewandte Mathematik (Crelles ...
Houa's user avatar
  • 561
3 votes
0 answers
214 views

Extended adjoint of Volterra operator

Let $V$ be a Volterra operator on $L^2 [0,1]$. Does there exist a nonzero operator $X $ satisfying the following system $VX=XV^∗$, where $V^∗$ is the adjoint of the Volterra operator? $$ V(f) (x) =\...
Anas Abbas H.'s user avatar
3 votes
0 answers
393 views

On a possible attempt to prove the invariant subspace problem

This question involves a possible method to prove the invariant subspace problem for (separable) infinite dimensional Hilbert spaces. The idea comes from various results on this topic; more precisely, ...
Manuel Norman's user avatar
3 votes
0 answers
68 views

A strange convergence for a semigroup of operators

I am reading B. Simon's "Kato's inequality and the comparison of semigroups", and I am having troubles understanding a part of the proof of Theorem 1 therein, that goes as follows: Let $A,B$ ...
Alex M.'s user avatar
  • 5,407
3 votes
0 answers
63 views

Continuity of local spectral radius

Let $H$ be a complex Hilbert space and let $T \in \mathcal{B}(H)$ be a linear, bounded operator. Given $x \in H$ we define its local spectral radius as $$r_T(x) = \limsup\limits_{n\rightarrow\infty} \|...
javi1996's user avatar
  • 355
3 votes
0 answers
225 views

Defining a trace-class operator with a Bochner integral

I had asked this question previously on Math.StacheExchange but did not get an answer there in several months. This isn't strictly speaking research level mathematics but I hope it is sufficiently ...
Adomas Baliuka's user avatar
2 votes
0 answers
71 views

How to naturally define an output space with certain properties

Consider the following regression problem $v=A(u) + \varepsilon$ for some operator $A:\mathcal{U} \rightarrow \mathcal{V}$ and some function spaces $\mathcal{U},\mathcal{V}$, mapping from $\mathcal{X}$...
emma bernd's user avatar
2 votes
0 answers
318 views

What are alternative or equivalent definitions of a positive-definite function on a group?

The standard definition of a positive-definite function on a group goes as follows: Let $\varphi : G \rightarrow L(H)$, where $G$ is a group (with an involution) and $H$ a Hilbert space. $L(H)$ is the ...
S-F's user avatar
  • 63
2 votes
0 answers
177 views

What are the current open problems in dilation theory?

I just started doing my PhD in mathematics. My topic is unitary dilations of operators. I've been reading a lot of papers on that subject so far (especially about the dilation of $n \ge 3$ commuting ...
S-F's user avatar
  • 63
2 votes
0 answers
77 views

Why is the essential numerical range defined as $W_e(T) = \bigcap_{K\in \mathcal K(H)} \overline{W(T+K)}$?

I have been introduced to the following definition of the essential numerical range of a bounded, linear operator on a separable, infinite-dimensional Hilbert space: $$W_e(T) := \bigcap_{K\in \mathcal ...
stoic-santiago's user avatar
2 votes
0 answers
56 views

Existence of a suitable smooth kernel

Denote by $H=L^2([0,1])$ the Hilbert space of square integrable real valued functions on the interval $[0,1]$. Does there exist a nontrivial smooth real valued function $k$ on the unit interval such ...
Ali's user avatar
  • 4,143
2 votes
0 answers
107 views

The density of the image of a unitary irrep (a generalization of Burnside's theorem)

I asked the following question on MSE and never got an answer. I am curious if there are any generalizations of Burnside's theorem (If $(\pi,V)$ is irreducible, then $\pi(G)$ spans $\operatorname{End}(...
Eric Kubischta's user avatar
2 votes
0 answers
73 views

A question on groupoids and measurable fields of Hilbert spaces

Suppose that we have the following data: $ \mathcal{G} $ is a locally compact Hausdorff groupoid, with its source and range maps denoted by $ s $ and $ r $ respectively. $ (\lambda^{x})_{x \in \...
Transcendental's user avatar
2 votes
0 answers
341 views

Trace class operators convergent series

On wikipedia it is mentioned that if we are on some (separable) Hilbert space $H$ and there is an ONB $(e_n)$ such that any compact operator $K$ can be written as $$ K = \sum_{n,m =0}^{\infty} K_{n,m}...
Kinzlin's user avatar
  • 305
2 votes
0 answers
97 views

essential self-adjointess for operators that can be factorized as $TT^*$

Let $X,Y$ be Hilbert spaces, $D$ be a dense subspace of $X$, $T:D\to Y$ be a linear operator, $\tilde{D}:=T(D)$. Assume $T^*T:D\to X$ to be essentially self-adjoint and the generated semigroup $(e^{-...
Delio Mugnolo's user avatar
2 votes
0 answers
51 views

Multivariable Weighted shift and subnormality

I have asked this question in mathstackexchange but didn't get any answer. I hope, I will get my answer here. Let $\mathbb B^m$ denote the Euclidean ball in $\mathbb C^m.$ Does there exist a ...
Timon's user avatar
  • 207
2 votes
0 answers
82 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\ \...
Michael Tinker's user avatar
1 vote
0 answers
87 views

Convergence and sequential compactness for nonlinear operators

I have a family of operators $T_n\colon X \to Y$ where $X,Y$ are Hilbert spaces. These operators are nonlinear. What kind of notions of convergence does one have for such operators? I'm specifically ...
C_Al's user avatar
  • 251
1 vote
0 answers
120 views

Formula for the kernel of an operator

Let $\mathcal H$ be a Hilbert space and let $O$ be an operator. Obviously $M=O^\dagger O$ is a semi-positive definite operator and $v\in\ker M$ if and only if $v\in\ker O$. Therefore it seems to me ...
dennis's user avatar
  • 521
1 vote
0 answers
89 views

Let $T\in B(H)$. Then prove that $T$ is a contraction if and only if the closed unit disc is spectral set for $T$

Let's first recall the definition of Spectral set for a bounded operator $T$ on a hilbert space $H$. Let $X\subseteq \Bbb{C}$ be compact such that $\sigma(T)\subseteq X$. Define $$\mathcal{R}(X):=\...
DeltaEpsilon's user avatar
1 vote
0 answers
33 views

Limiting absorption principle for higher powers of resolvents

Let $H$, $A$ be self-adjoint operators on a Hilbert space. Moreover, let $I$ be a bounded open interval contained in the spectrum of $H$. Assume that $H$,$A$ satisfy the following positive commutator ...
Janik's user avatar
  • 141
1 vote
0 answers
122 views

eigenvalues of integral operator with centered kernel

Suppose $k:\mathcal{X} \times \mathcal{X} \to \mathbb{R}$ be a symmetric positive (semi-)definite kernel. The Moore-Aronszajn Theorem indicates that there is a reproducing kernel Hilbert Space $\...
Kcafe's user avatar
  • 519
1 vote
0 answers
26 views

On some bounds on two constants concerning the disconnectedness of the spectra of small perturbations of operators

Let $H$ be a separable, infinite dimensional, complex Hilbert space. In the book: Jiang, C. L.; Wang, Z. Y. (1998). Strongly Irreducible Operators on Hilbert Space. CRC press above the statement of ...
Manuel Norman's user avatar
1 vote
0 answers
70 views

Is $F: B(H)\to B(H\otimes K)$, $X\mapsto X\otimes \mathbb{1}_K$ completely contractive?

Take Hilbert spaces $H$ and $K$. Consider a linear map $F: B(H)\to B(H\otimes K)$, $X\mapsto X\otimes \mathbb{1}_K$. Is it true that $F$ is completely contractive? If it is, I would be very grateful ...
Maria  Dmitrieva's user avatar
1 vote
0 answers
80 views

Measurability of a generalized point spectrum

Assume that $ T:H\oplus H\rightarrow H\oplus H$ is a unitary linear operator on the double sum of a separable Hilbert space $H$ with itself. Let us call a pair $(\lambda, \mu)\in\mathbb{C}\oplus\...
Dmitri Scheglov's user avatar
1 vote
0 answers
60 views

Is there a vector-valued trace such that $\text{tr}((L\otimes_π\text{id}_H)T)=LT$ for all $L∈\mathfrak L(H,\mathfrak L(H))$ and $T∈H\hat\otimes_πH$?

Let $H$ be a separable $\mathbb R$-Hilbert space $L\in\mathfrak L(H,\mathfrak L(H,\mathbb R))$ $T\in\mathfrak L(H)$ be nonnegative, self-adjoint and nuclear (trace-class) Note that$^1$ $$\...
0xbadf00d's user avatar
  • 167
1 vote
0 answers
127 views

A point in Ion Suciu's paper on semigroups of isometric operators

My question is concerned a point in this 1968 paper by Ion Suciu which is given in Theorem 2. In the last paragraph of page 104, it is claimed that $N$ (given in the formula 2.5) is a wandering ...
ABB's user avatar
  • 4,058
1 vote
0 answers
233 views

Bochner integrals with values in a Hilbert $A$-module

I'm wondering whether there exists a generalisation of Bochner integration with values in a Hilbert $A$-module $M$, where $A$ is a general $C^*$-algebra rather than $\mathbb{C}$ (and whether there are ...
geometricK's user avatar
  • 1,903
1 vote
0 answers
220 views

About the projection on the unit sphere

Let $H$ be a Hilbert Space and let $A\subset H$ be a connected set such that any two elements of $A$ are linearly independent and also $A^{\bot}=\left\{0\right\}$ (this seems to be immaterial). Is ...
erz's user avatar
  • 5,529
1 vote
0 answers
122 views

Algorithm for finding eigenfunctions

I have an $ L^2(\mathbb{R}) $ operator that looks like $$ \Omega = \int \partial\phi(a, b)\ \ |b, a\rangle \langle b, a |, $$ where $ \langle x | a, b \rangle = f_a(x - b) e^{x^2/2} $ and $ f_a \in L^...
user avatar
0 votes
0 answers
35 views

Operator-form correspondence without lower semiboundedness

When dealing with a normal unbounded operator $A$, it is often useful to frame questions about the operator in terms of questions about the associated form $\omega,$ which has domain $D(|A|^{1/2})$ ...
user_35's user avatar
  • 109
0 votes
0 answers
72 views

Domain of a Jacobi operator with unbounded coefficients

Is it possible to describe the domain of a Jacobi operator explicitly? Let $J$ be the linear operator acting on a real sequence $(u_{n})_{n\in\mathbb{N}}$ by $$ J(u_{n}) = a_{n+1} u_{n+1} + a_{n} u_{n-...
Bastien's user avatar
  • 23
0 votes
0 answers
138 views

Question about a step in the proof of the min-max principle

I honestly do not think this is a hard question, maybe it is even obvious but I tried MSE and had no success so far, so I am reproducing the question Question about the proof of the min-max principle ...
MathMath's user avatar
  • 1,305
0 votes
0 answers
63 views

Inequality for normed power n, m

Let $ B (H) $ indicate the set of all bounded linear operators on a complex separable Hilbert space $ H $. Let $ A \in B(H) $, where $ A $ is a positive semi-definite operator in $ H $ (i.e. $ \langle ...
Anas Abbas H.'s user avatar
0 votes
0 answers
213 views

Convergence of inverse operator with projections

Let $X$ be a separable Hilbert space, and let $(e_i)_{i=1}^\infty$ be an orthonormal basis of $X$. For each $n\in \mathbb{N}$, let $X_n$ be the subspace spanned by $(e_i)_{i=1}^n$, and consider the ...
John's user avatar
  • 503
0 votes
0 answers
69 views

Explicit description for dual to operator space of Hilbert space

Let $H$ be a separable Hilbert space, and $B := \mathcal B(H)$ be the space of bounded operators on $H$. It is known that $B^\ast_\mathrm{strong} = B^\ast_\mathrm{weak}$ (see [Dunford, Schwartz, VI.1....
Vasily Ionin's user avatar
0 votes
0 answers
122 views

Isolated points of the spectra of self-adjoint operators on Hilbert spaces

Let $T$ be a (everywhere defined) self-adjoint operator on a complex Hilbert space $\mathcal{H}$. I am interested in results that give (non-trivial, possibly mild) sufficient conditions on $T$ to ...
Maurizio Moreschi's user avatar
0 votes
0 answers
255 views

The limit of the operator norm in a Hilbert space

I am not familiar with functional analysis. Could you tell me please, how to prove the following statement (if it is true)? $$ \lim_\limits{M \to \infty} \|T_A - T_b \| = 0, $$ here operator norm ...
MightyPower's user avatar
0 votes
0 answers
55 views

Dense stratification of a separable Hilbert space

Let $\{X_i\}_{i \in \mathbb{N}} $ be a sequence of $n$-dimensional linear subspaces of the separable Hilbert space $H$ and let $\{\phi_i\}_{i \in I}$ be a sequence of continuous injective linear maps ...
ABIM's user avatar
  • 5,405
0 votes
0 answers
87 views

Orthogonal functions and linear operators

Consider the following function $f: [-1,1] \rightarrow \mathbb{R}$, expanded in terms of Legendre functions, $$ f(y;\boldsymbol{\beta}) = \sum_{i=0}^{\infty} \beta_i P_i(y) $$ where $\boldsymbol{\beta}...
user3516849's user avatar
0 votes
0 answers
87 views

Uniform convergence in Hadamard derivatives

Let $T\colon X \to Y$ be a nonlinear operator between Hilbert spaces which is Lipschitz and is Hadamard differentiable. It satisfies $$T(x+th)=T(x) + tT'(x)(h) + r(t)$$ where $r(t)=r(t,x,h)$ is the ...
M.L's user avatar
  • 73