# Questions tagged [hilbert-spaces]

A Hilbert space $H$ is a real or complex vector space endowed with an inner product such that $H$ is a complete metric space when endowed with the norm induced by this inner product.

660 questions
Filter by
Sorted by
Tagged with
34 views

42 views

1 vote
74 views

70 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 ...
87 views

70 views

1 vote
35 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 $\psi_h(x)$...
45 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 ...
1 vote
113 views

### The space of linear operators between Hilbert spaces has martingale type 2

I am trying to prove whether the space $L(H,K)$ has martingale type 2 for Hilbert spaces $H,K$. It is known that Hilbert spaces have martingale type 2, so I was wondering whether the space of bounded ...
23 views

128 views

### How to prove $\|I-P\| = \|P\|$ for any non-trivial projector? [duplicate]

I noticed that in the paper  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 ...
1 vote
56 views

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} ... 7 votes 0 answers 117 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 ... 1 vote 1 answer 139 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 vote 0 answers 74 views ### Invariance signature in infinite dimension Let V be an infinite dimensional vector space and suppose we have a smooth family \{g_t\}_{t\ge 0} of symmetric bi-linear forms such that: g_0 is positive-definite g_t is non-degenerate for ... 0 votes 0 answers 77 views ### Operators decomposition in pseudo-Hilbert space Let (H,g) be a pseudo-Hilbert space, i.e. H is an infinite dimensional vector space endowed with an indefinite symmetric product g. Suppose we have a linear operator D:H\to H and let D^* be ... 2 votes 0 answers 114 views ### What are examples of infinite-dimensional Banach spaces that are also measure spaces? I am interested in examples of infinite-dimensional vector spaces that are Banach spaces or even Hilbert spaces are measure spaces Instead of the full vector space, subsets with measure structure ... 1 vote 1 answer 73 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 ... 7 votes 2 answers 292 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 \... 4 votes 1 answer 478 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 ... 0 votes 0 answers 112 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 ... 3 votes 0 answers 111 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 ... 7 votes 1 answer 173 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 ... 3 votes 1 answer 313 views ### Motivation for Heisenberg's modeling of observables What's the motivation for observables to be modeled by self-adjoint operators? I can't seem to find any place where this is laid out clearly. Maybe von Neumann's book is decent, but it's not ... 3 votes 3 answers 300 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 (... 4 votes 1 answer 106 views ### Is the Sobolev space H^1(\mathbb{R}) contained in the domain of (-\partial_x \alpha(x) \partial_x)^{1/2}? Let \alpha(x) : \mathbb{R} \to (0,\infty) have bounded variation (BV) and suppose \inf_{\mathbb{R}} \alpha > 0. Consider the second order differential operator$$H : =-\partial_x (\alpha(x) \...
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 $V:R(A)\to \overline{R(T)}$ ...
### 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....