All Questions
467 questions
1
vote
1
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355
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Hilbert–Pólya conjecture with Grommer inequalities?
The Grommer inequalities are equivalent to RH and formulated on page 20 of Conrey - Riemann's hypothesis:
Let
$$\Xi(t) := \xi(1/2+it).$$
Then RH is equivalent to : All zeros of $\Xi(t)$ are real. The ...
- 3,064
1
vote
0
answers
82
views
Commutator of self-adjoint operators and $C^1$-type formula
Let $\mathcal{H}$ be a (complex) Hilbert space.
Let $H$ be a self-adjoint operator on $\mathcal{H}$ with dense domain $\mathcal{D}(H) \subset \mathcal{H}$, generating the unitary one-parameter ...
- 71
0
votes
1
answer
218
views
Intersection of Hilbert spaces with Schauder basis
Let
$H$ be a infinite dimensional, separable, complex Hilbert space,
$\{v_{1_n}\}_{n \in \mathbb{N}}$ be a sequence in $H$,
$V_1=\operatorname{span}\{v_{1_n}\}_{n \in \mathbb{N}}$
$U_1=\overline{V_1}$...
- 433
2
votes
2
answers
167
views
LF or LB space that happens to be finite dimensional
Let $\{V_n\}_{n=1}^\infty$ be a collection of finite dimensional vector subspaces of $L^2[0,1]$ such that $V_n \subset V_{n+1}$ and $\bigcup_{n=1}^\infty V_n$ is dense in $L^2[0,1]$. Suppose further ...
- 3,477
2
votes
0
answers
258
views
Orthogonal complement of arbitrary intersection of Hilbert subspaces
Let $H$ a Hilbert space, and $\mathcal C$ an arbitrary set of closed subspaces of $H$. Is it true that
$$\left( \bigcap_{Z\in \mathcal C}Z\right)^\perp = \overline{\sum_{Z\in \mathcal C} Z^\perp}$$
...
- 63
0
votes
0
answers
63
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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 ...
1
vote
1
answer
354
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Mach's principle, Newton's law and Hilbert sphere?
(This question has originally been posted on reddit, but I thought, that the question raised in the post above, might fit as well here on MO.)
I wanted to share with you something I stumbled upon ...
- 3,064
-4
votes
2
answers
530
views
Inverse square-law as a positive definite kernel?
Newtons law for gravity states that:
$$F_{12} = \frac{G m_1 m_2} {|x_1-x_2|^2}$$
The function :
$$k(x,y):=\exp(-| x-y|^2)$$
is known to be a positive definite function, called the RBF-kernel.
It ...
- 3,064
0
votes
1
answer
119
views
A property of the canonical dual frame in a Hilbert space
Let $\{ g_n \} $ be a frame in a separable Hilbert space $H$. Then the frame operator $S:H\to H$ defined as
\begin{equation} S f := \sum_{n=1}^\infty (f,g_n)g_n \end{equation}
is a Hilbert space ...
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
$${\rm Tr}(l(a))={\rm Tr}(r(a))<\infty , {\rm Tr}(l(b))={\rm Tr}(r(b))<\infty,$$
$${\rm Tr}(l(a)+l(b)) ={\rm Tr}(l(\alpha_1 a +\...
- 617
3
votes
1
answer
209
views
Must solutions to the time-independent Schrodinger equation that have discrete or negative eigenvalues be square-integrable?
This is a very straightforward question (although the answer may not be!) about a ubiquitous textbook physics situation. I assume that the answer is probably well-known, but I asked it on both Math ...
- 1,311
0
votes
0
answers
213
views
Convergence of inverse operator with projections
Let $X$ be a separable Hilbert space, and let $(e_i)_{i=1}^\infty$ be an orthonormal basis of $X$. For each $n\in \mathbb{N}$, let $X_n$ be the subspace spanned by $(e_i)_{i=1}^n$, and consider the ...
- 503
1
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0
answers
77
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Integration over a finite-dimensional subspace of Hilbert space
Let $H$ be a separable Hilbert space with inner product $\langle,\rangle$, let $\{e_k\}_{k=1}^\infty$ be an orthonormal basis of $H$, and let $A: H\to H$ be a symmetric, positive definite and ...
- 503
1
vote
0
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89
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Let $T\in B(H)$. Then prove that $T$ is a contraction if and only if the closed unit disc is spectral set for $T$
Let's first recall the definition of Spectral set for a bounded operator $T$ on a hilbert space $H$. Let $X\subseteq \Bbb{C}$ be compact such that $\sigma(T)\subseteq X$. Define $$\mathcal{R}(X):=\...
- 111
4
votes
0
answers
152
views
Maximally fine topologies on $B(H)$ making the unit ball compact
Let $H$ be a Hilbert space, and $B(H)$ its algebra of bounded operators. One of the reasons the Ultraweak topology is (in a way) more useful than the weak operator topology is that the Ultraweak ...
- 190
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
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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
$$
\lVert p(T)\rVert \leq \sup \...
- 31
1
vote
0
answers
62
views
$L^2$ norm of a kernel with a variable width
Suppose you have a kernel operator on a torus, with a kernel of a spatially varying width $\epsilon(x)$, which might be zero at certain points. That is to say, for some approximate identity $\psi_h(x)$...
2
votes
0
answers
73
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RKHS lying in another RKHS
Suppose $H_1$ and $H_2$ are reproducing kernel Hilbert spaces such that $H_1 \subset H_2$. For $f \in H_1$, when can I bound $\|f \|_1$ with $C\|f\|_2$ (for some $C$)? Is there a relationship between ...
- 93
2
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0
answers
134
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How to prove $\|I-P\| = \|P\|$ for any non-trivial projector? [duplicate]
I noticed that in the paper [1] this property is proved by explicitly computing the singular values of $P$ after realizing $P$ in finite dimension as oblique projection matrix. I am wondering if there ...
- 205
1
vote
1
answer
89
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Show $A_0 + S$ is invertible, where $S$ is the Riesz projection of bounded $A_0$
Let $A_0$ be a bounded linear operator on a Hilbert space $H$. Suppose $0$ is an isolated point of the spectrum of $A_0$. Let $S$ be the corresponding Riesz projection, namely,
$$S = -\frac{1}{2\pi i} ...
- 481
6
votes
0
answers
188
views
Measurability of eigenvalues-eigenvectors of a positive compact operator
Let $H$ be a separable Hilbert space over $\mathbb{R}$. Let ${A} = \{a\colon H\to H\,|\,a\text{ is a positive, compact linear operator}\}$.
By the spectral theorem, given $a \in A$, there are scalars $...
- 163
1
vote
1
answer
299
views
Isometries of Hilbert space
It is easy to see that for any $x$ and $y$ on the unit sphere of a Hilbert space $H$ there exists a surjective isometry $U$ such that $Ux=y$. Does something more general also hold? That is, given two ...
- 1,361
1
vote
1
answer
89
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Representing an $L^2$-functional by a non-$L^2$-function on a dense subspace - Part II
This is a follow-up to this previous question, but under stronger assumptions.
Let $(X, \mu)$ be a (say, $\sigma$-finite) measure space, let $g \in L^2$ (say, over the real
scalar field). Let $\tilde ...
- 12.6k
7
votes
2
answers
464
views
Representing an $L^2$-functional by a non-$L^2$-function on a dense subspace
Let $(X, \mu)$ be your favourite measure space (finite or $\sigma$-finite if you like), let $g \in L^2$ (say, the scalar field of $L^2$ is $\mathbb{R}$, though this probably doesn't matter). Let $\...
- 12.6k
5
votes
1
answer
1k
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Left and right eigenvectors are not orthogonal
Consider a compact operator $T$ on a Hilbert space with algebraically simple eigenvalue $\lambda$. Is it then true that left (the eigenvector of the adjoint with complex-conjugate eigenvalue) and ...
- 73
0
votes
0
answers
220
views
Eigenvalue multiplicity of tensor product of positive operator with itself
Let $H$ be a separable complex Hilbert space and let $A\in B(H)$ be positive with $||A||=1$ and have eigenvalue 1 with multiplicity 1. Suppose $A=T^*T$ for some $T\in B(H)$. Denote the spectrum of $A$ ...
- 203
3
votes
0
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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
3
votes
3
answers
319
views
Do these properties characterize Hilbert spaces?
Suppose $X$ is a Banach space with the following property: For any $x\in X$ there exists a two dimensional subspace $E$ isometric with $l_2^2$ such that $x\in E$. Does this property characterize a (...
- 1,361
3
votes
0
answers
390
views
How to prove the polar decomposition of unbounded operators?
Let $ T $ be a closed, densely defined operator on a Hilbert space $ H $. Then there exists a positive self-adjoint operator $ A $, $ D(A)=D(T) $ and a isometric operator $ V:R(A)\to \overline{R(T)} $ ...
- 1,219
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
2
votes
1
answer
154
views
Function monotony between [0,T] and $L^2$
Let $\Omega\subseteq\mathbb{R}^N$ be a bounded and smooth domain. If $z:[0,T]\to L^2(\Omega)$ is a function in $H^1([0,T],L^2(\Omega))$ with the property that $z'(t)(x):=z'(t,x)>0$ a.e. on $\Omega$ ...
- 1,759
9
votes
2
answers
516
views
Why operator systems?
A $\mathrm{C}^*$-algebra $\mathcal{A}\subset B(\mathsf{H})$ is a norm-closed, self-adjoint subalgebra of bounded operators on a Hilbert space. If we then take a unital self-adjoint (possibly closed) ...
- 1,037
2
votes
0
answers
77
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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:
$$W_e(T) := \bigcap_{K\in \mathcal ...
- 165
2
votes
0
answers
56
views
Existence of a suitable smooth kernel
Denote by $H=L^2([0,1])$ the Hilbert space of square integrable real valued functions on the interval $[0,1]$. Does there exist a nontrivial smooth real valued function $k$ on the unit interval such ...
- 4,143
1
vote
0
answers
99
views
Density of Lipschitz functions in Bochner space with bounded support
Let $X$ and $Y$ be separable and reflexive Banach spaces with Schauder bases. Let $\mu$ be a non-zero finite Borel measure on $X$ and let $L^p(X,Y;\mu)$ denote the (Boehner) space of strongly p-...
- 21
5
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0
answers
116
views
Multiplier algebra of Fock space
For any vector space, one may form the tensor algebra with multiplication being the tensor product. For a Hilbert space $\mathcal{H}$, the analogous construction is the Fock space
$$
\mathcal{F}(\...
- 439
5
votes
1
answer
386
views
Contact points for John's ellipsoid
Suppose $K$ is a centrally symmetric convex body in $\mathbb{R}^n$ and $E$ is the John's ellipsoid, the ellipsoid of maximal volume inside $K$.
If $E$ and $K$ have exactly $2n$ contact points, say $(\...
- 1,361
0
votes
1
answer
278
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A reproducing kernel Hilbert space
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
8
votes
1
answer
446
views
Parallelogram law for vectors of equal length
Does the parallelogram law for vectors of equal length imply the full parallelogram law? That is,
if for all norm one vectors $x$ and $y$ in a Banach space $X$ it holds that $\lVert x-y\rVert^2+\lVert ...
- 1,361
0
votes
1
answer
269
views
Determine if an integral expression is in $L^2(\mathbb{R})$
Note: This is a simplified version of the following question. I did not get a full response and realized can make it simpler to have my main interrogation answered. I decided to write it as a ...
0
votes
0
answers
182
views
Comparison of orthogonal complements in L^2 and H^1 spaces
Let $\Omega$ be a bounded domain with smooth boundary, and let $L^2(\Omega)$ and $H^1(\Omega)$ the usual $L^2$ and $H^1$ function spaces on $\Omega$, respectively. We call $\phi \in H^1(\Omega)$, and $...
- 11
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
-1
votes
1
answer
164
views
Closure of the point spectrum of an unbounded diagonalizable operator
Given a (separable) Hilbert space H and an unbounded densely defined linear operator $T:{\cal D}(T) \to $H such that ${\cal D}$ is diagonalizable (it means $\exists$ an O.N.B. of H such that all basis ...
- 969
1
vote
0
answers
83
views
What is lost after RKHS embedding of the L1 space?
We know that every distribution or $L^1$ function $f$ over space $\mathcal{X}$ (e.g., $R^d$) can be embedded to an RKHS $\mathcal{H}$ with a $1$-bounded kernel $\mathcal{K}$ (e.g., the RBF kernel) ...
- 43
1
vote
0
answers
111
views
Properties of Sobolev spaces $W^{k,p}(\Omega, E)$ where $E$ is a Banach space
$\newcommand{\R}{\mathbb R}$Let $E$ be a Banach space with norm $\|\cdot\|_E$ and let $\Omega\subset \R^n$ be an open set.
For $k\geq 0, p\geq 1$ we define $W^{k,p}(\Omega, E)$, the Sobolev space of ...
- 2,533
6
votes
0
answers
529
views
Infinite-dimensional "algebraic varieties"
This question was also formerly posted on MSE but has not received any answer or comment.
Let $H$ be the infinite-dimensional seperable complex Hilbert space, and $P(H)$ denote its projectivization. ...
- 1,543
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