All Questions
Tagged with hilbert-spaces operator-theory
161 questions
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 ...
- 31
5
votes
2
answers
149
views
Showing an operator is (or not) closed on $L^2(\mathbb{R})$
I am linearizing nonlinear waves and get operators of the form below. Everything is considered in $L^2(\mathbb{R})$.
Consider the operator $L_1=\frac{d}{dx}$. The domain is $H^1(\mathbb{R})$ and it is ...
7
votes
1
answer
957
views
a claim for a proof of the invariant subspace problem [closed]
Recently four mathematicians claimed to have proven the invariant subspace problem, which is the problem that states
Does every bounded operator on a separable Hilbert space have a non-trivial ...
- 87
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 ...
- 251
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 $D(|A|^{1/2})$ ...
- 109
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 $T$...
- 233
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 $\mathcal{X}$...
- 51
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
$$
J(u_{n}) = a_{n+1} u_{n+1} + a_{n} u_{n-...
- 23
-3
votes
1
answer
76
views
Minimal norm problem with linear combination of translation operator to be estimated
Follow up question from this one
Suppose $X = L^2(G)$, where $G$ is some locally compact group. Let $x, y \in G$ I for fixed $n$ I am seeking for an operator $H \in B(X)$ of the form
$$
H = H(\alpha_1,...
- 115
2
votes
1
answer
205
views
$L^2$ space of Hilbert-Schmidt operator valued functions
Let $\mathscr{S}$ denote the space of all Hilbert-Schmidt operators on $L^2(\mathbb R)$. Consider the Hilbert space $L^2(\mathbb R, \mathscr S)$ of square-integrable $\mathscr S$-valued function, that ...
- 407
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 ...
- 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 ...
- 63
2
votes
1
answer
237
views
On spectral calculus and commutation of operators
Let $\mathcal{H}$ be a Hilbert space, $B\in\mathcal{B}(\mathcal{H})$ be bounded and self-adjoint and $A:\mathcal{D}(A)\to\mathcal{H}$ closed (but not necessarily self-adjoint or bounded). The ...
- 1,171
1
vote
1
answer
44
views
Wold decomposition of toral endomorphisms
Suppose that $A\in M_d(\mathbb{Z})$ is a $d \times d$ matrix with non zero determinant and suppose that $\mathbb{T}^d$ is the $d$-dimensional torus. Then one can define an operator on $L^2(\mathbb{T}^...
1
vote
1
answer
217
views
Perturbation of matrices
Let $A(t)$ be a symmetric $n\times n$ matrix that continuously depend on $t\in [0,1]$. Let $\lambda_1(t)$ stand for the smallest eigenvalue for $A(t)$.
Question. Does there exist a Lebesgue measurable ...
- 4,135
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 ...
- 1,305
0
votes
1
answer
184
views
Spectrum of a product of a symmetric positive definite matrix and a positive definite operator
Let $\mathbf H$ be an infinite dimensional Hilbert space.
I want to find an example of a $2\times 2$ real symmetric positive definite matrix $M$ and a positive definite bounded operator $A : \mathbf H ...
- 63
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 ...
- 521
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 ...
0
votes
1
answer
93
views
Does ${\rm Tr}(l(a)+l(b)) ={\rm Tr}(l(\alpha_1 a +\alpha_2 b ))$ imply that $l(a)l(b)=r(a)r(b)=0$?
Let $H$ be a Hilbert space and let $a,b\in B(H)$ be such that
$${\rm Tr}(l(a))={\rm Tr}(r(a))<\infty , {\rm Tr}(l(b))={\rm Tr}(r(b))<\infty,$$
$${\rm Tr}(l(a)+l(b)) ={\rm Tr}(l(\alpha_1 a +\...
- 617
0
votes
0
answers
213
views
Convergence of inverse operator with projections
Let $X$ be a separable Hilbert space, and let $(e_i)_{i=1}^\infty$ be an orthonormal basis of $X$. For each $n\in \mathbb{N}$, let $X_n$ be the subspace spanned by $(e_i)_{i=1}^n$, and consider the ...
- 503
2
votes
1
answer
245
views
If $A$ is a closed operator, is $A^k$ closed?
Let $A$ be a closed (densely defined) operator on a Hilbert space $H$.
We define for a natural number $k$, the operator $A^k$ with its natural domain.
Is $A^k$ closed?
- 175
1
vote
0
answers
89
views
Let $T\in B(H)$. Then prove that $T$ is a contraction if and only if the closed unit disc is spectral set for $T$
Let's first recall the definition of Spectral set for a bounded operator $T$ on a hilbert space $H$. Let $X\subseteq \Bbb{C}$ be compact such that $\sigma(T)\subseteq X$. Define $$\mathcal{R}(X):=\...
- 111
2
votes
1
answer
233
views
Identity for spectral resolution: $dE_{\xi, \xi}= |g|^2 dE_{\eta, \eta}$
Let $(\Omega, \mathcal{F})$ be a measurable space. Let $E: \mathcal{F}\to B(H)$ be a regular resolution of the identity on the Hilbert space $H$, see e.g. Rudin's functional analysis book.
Suppose ...
- 175
3
votes
1
answer
332
views
Takesaki II Lemma 1.13: stuck in proof
Consider the following fragment from Takesaki's book "Theory of operator algebras II" (Lemma 1.13 on p8, in chapter VI "Left Hilbert algebras"):
Here, we associate with an ...
- 175
5
votes
2
answers
276
views
Dilation of bounded linear operators
Let $H$ be a Hilbert space, and let $A$ be a contraction (bounded linear operator of norm $\leq 1$) on $H$. I heard in a recent talk that there is a (apparently famous) result due to Sz-Nagy which ...
- 51
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
$$
\lVert p(T)\rVert \leq \sup \...
- 31
3
votes
0
answers
122
views
Domain of operator which is used in operator monotone function
We are studying the paper
Rupert L. Frank, Leander Geisinger, Refined semiclassical asymptotics for fractional powers of the Laplace operator, Journal für die reine und angewandte Mathematik (Crelles ...
- 561
7
votes
1
answer
283
views
A characterization of Hilbert spaces by norm one projections
Suppose a (separable) Banach space $X$ has the following property: If $P:X\to X$ is a bounded projection different from $I$ such that $\|P\|=1$, then $\|I-P\|=1$. Does this imply that $X$ is a Hilbert ...
- 1,361
1
vote
1
answer
60
views
Is $\sup\{\|A\widehat{k}_{\lambda}\|: \lambda\in\Omega\}=\sup\{\|A^*\widehat{k}_{\lambda}\|: \lambda\in\Omega\}$? where $A$ is an operator on RKHS
A functional Hilbert space $\mathscr H=\mathscr H(\Omega)$ is a Hilbert space of complex valued functions on a (nonempty) set $\Omega$, which has the property that point evaluations are continuous i.e....
- 1,154
0
votes
0
answers
69
views
Explicit description for dual to operator space of Hilbert space
Let $H$ be a separable Hilbert space, and $B := \mathcal B(H)$ be the space of bounded operators on $H$.
It is known that $B^\ast_\mathrm{strong} = B^\ast_\mathrm{weak}$ (see [Dunford, Schwartz, VI.1....
- 101
2
votes
0
answers
77
views
Why is the essential numerical range defined as $W_e(T) = \bigcap_{K\in \mathcal K(H)} \overline{W(T+K)}$?
I have been introduced to the following definition of the essential numerical range of a bounded, linear operator on a separable, infinite-dimensional Hilbert space:
$$W_e(T) := \bigcap_{K\in \mathcal ...
- 165
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 ...
- 4,135
3
votes
1
answer
475
views
Extension of a bounded linear functional
Let $\mathcal{H}$ be a finite-dimensional Hilbert space and $\mathcal{A}\subseteq\mathcal{B(\mathcal{H})}$ be an operator system. Suppose $T_n$ is a collection of all $n$-by-$n$ matrices equipped with ...
- 231
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 ...
- 141
0
votes
1
answer
130
views
Diagonaloperators on $l^2(N)$ [closed]
Let $D$ be an operator defined by $D(c_n)_{n\in\mathbb{N}} = (a_n c_n)_{n\in\mathbb{N}}$. It's a well known fact that $D$ is well defined as an operator from $l^2(\mathbb{N})$ to $l^2(\mathbb{N})$ if $...
- 3
3
votes
2
answers
135
views
Unicellular compact operators
An operator $T$ on a separable Hilbert space $H$ is called unicellular if any two closed invariant subspaces $M$ and $N$ are comparable; that is either $M\subseteq N$ or $N\subseteq M$. There are many ...
- 1,361
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?
$$ V(f) (x) =\...
2
votes
0
answers
107
views
The density of the image of a unitary irrep (a generalization of Burnside's theorem)
I asked the following question on MSE and never got an answer.
I am curious if there are any generalizations of Burnside's theorem (If $(\pi,V)$ is irreducible, then $\pi(G)$ spans $\operatorname{End}(...
- 161
0
votes
1
answer
138
views
Antilinear unbounded operator has closed graph
Let $H$ and $K$ be Hilbert spaces and $D(T)$ a vector subspace of $H$. Let $T: D(T) \to K$ be a densely defined antilinear operator. Its adjoint $T^*: D(T^*)\to K$ is defined by the relation
$$\langle ...
- 175
1
vote
1
answer
143
views
A question on the self-adjointness of an operator
Given a Hilbert space (separable) $\mathcal{H}$ with an orthonormal basis $\{e_i\}_{i=1}^{\infty}$, define an operator $T$ with domain $\mathcal{D}(T)$ equal to the span of $\{e_i\}$ by $Te_i:=\...
user471371
3
votes
1
answer
140
views
Infinite-dimensional analogue of "positive-negative splitting implies non-degeneracy"
(This question is related to Splitting a space into positive and negative parts but different.)
Given a finite-dimensional vector space $V$ over $\mathbb{R}$, what I call a "positive-negative ...
- 1,804
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 $\...
- 519
20
votes
0
answers
333
views
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 ...
- 321
7
votes
3
answers
881
views
Criterion for compactness
Let $T:H\to H$ be a continuous operator on a Hilbert space.
Assume there exists an orthonormal base $(e_j)_{j\in\mathbb N}$, such that the sequence $Te_j$ tends to zero.
Must $T$ be compact?
user130903
3
votes
1
answer
261
views
norm estimates for Schatten class
Let $C
_p$ be the Schatten-p-classes on a separable Hilbert spaces, $p\ge 1$.
Let ${\rm Tr}$ be the standard trace.
Let $y\in C_p$ be a self-adjoint operator (or even a positive operator) and let $...
- 617
1
vote
1
answer
335
views
A consequence of the Min-Max Principle for self-adjoint operators
Let $H=(H, (\cdot, \cdot))$ be a Hilbert space. Let $T_1,T_2:D \subset H \longrightarrow H$ be a self-adjoint operators (not necessarily bounded). It's well-know that the spectrum $\sigma(T_i)$ of $...
- 205
0
votes
1
answer
102
views
Essential spectrum of constant invertible diagonal matrix acting on a product of Hilbert spaces [closed]
Let $M$ be a $3\times 3$ real invertible diagonal matrix and $H$ a Hilbert space of infinite dimension (for example, we can take $H$ as the space of square integrable functions over a bounded ...
- 63
6
votes
2
answers
514
views
Convergence criterion in the domain of an unbounded operator
Cross-post from math.sx.
My question is somewhat close to this one, but the counterexamples given there do not apply here.
Setup. Given a Hilbert space $\mathcal H$, a closed operator $A$ and a ...
- 245
3
votes
1
answer
157
views
Operator in the commutant which is small on a given vector
Suppose $x$ is a non-zero vector in a Banach space, and $T$ is a fixed operator. Is the following true:
For any $\varepsilon, \delta$, there exists $S$ in the commutant of $T$ such that $1\leq\|S\|<...
- 1,361