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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 ...
mathoverflowUser's user avatar
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 ...
DerGalaxy's user avatar
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 $H$, $V_1=\operatorname{span}\{v_{1_n}\}_{n \in \mathbb{N}}$ $U_1=\overline{V_1}$...
Matey Math's user avatar
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 ...
Isaac's user avatar
  • 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 $$\left( \bigcap_{Z\in \mathcal C}Z\right)^\perp = \overline{\sum_{Z\in \mathcal C} Z^\perp}$$ ...
Nathaël's user avatar
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
1 vote
1 answer
354 views

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 ...
mathoverflowUser's user avatar
-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 ...
mathoverflowUser's user avatar
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 ...
an_ordinary_mathematician's user avatar
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 +\...
user92646's user avatar
  • 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 ...
tparker's user avatar
  • 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 ...
John's user avatar
  • 503
1 vote
0 answers
77 views

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 ...
John's user avatar
  • 503
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
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 ...
Aareyan Manzoor's user avatar
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 ...
SKNEE's user avatar
  • 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 \...
HaSa's user avatar
  • 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 $\psi_h(x)$...
Caroline Wormell's user avatar
2 votes
0 answers
73 views

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 ...
Athere's user avatar
  • 93
2 votes
0 answers
134 views

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 ...
bernard's user avatar
  • 205
1 vote
1 answer
89 views

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} ...
JZS's user avatar
  • 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 $...
user127022's user avatar
1 vote
1 answer
298 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 ...
Markus's user avatar
  • 1,361
1 vote
1 answer
89 views

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 ...
Jochen Glueck's user avatar
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 $\...
Jochen Glueck's user avatar
5 votes
1 answer
1k views

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 ...
Guido Li's user avatar
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 $A$ ...
Dasherman's user avatar
  • 203
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
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 ...
Markus's user avatar
  • 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 (...
Markus's user avatar
  • 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)} $ ...
Luis Yanka Annalisc's user avatar
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....
Student's user avatar
  • 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 $\Omega$ ...
Bogdan's user avatar
  • 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) ...
JP McCarthy's user avatar
  • 1,027
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
  • 4,135
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-...
Wilson's user avatar
  • 21
5 votes
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}(\...
J_P's user avatar
  • 439
5 votes
1 answer
385 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 $(\...
Markus's user avatar
  • 1,361
0 votes
1 answer
278 views

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....
Student's user avatar
  • 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 ...
Markus's user avatar
  • 1,361
0 votes
1 answer
267 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 ...
Gateau au fromage's user avatar
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 $...
kichr's user avatar
  • 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 ...
Piku's user avatar
  • 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 ...
Janik's user avatar
  • 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 ...
Dave Shulman's user avatar
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) ...
epsilon's user avatar
  • 43
1 vote
0 answers
110 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 ...
Overflowian's user avatar
  • 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. ...
Zerox's user avatar
  • 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 ...
Markus's user avatar
  • 1,361

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