The hilbert-spaces tag has no wiki summary.
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Recontruction of the weak topolgy from the scalar product on a subset of a Hilbert Space
Let $M$ be a set a let $K:M\times M\to\mathbb{C}$ be a positive definite kernel. By a version of Moore-Aronszajn Theorem, there is a unique (up to the unitary euivalence) Hilbert Space $X$, and a map ...
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Hilbert scales of covariance operators
Assume we have 2 covariance operators(positive definite trace class) $S$ and $T$ on Hilbert space $\mathcal H$ with corresponding eigenpairs $\{e_j,\lambda_j\}$ and $\{f_j,\lambda_j\}$. Assume that
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57 views
strong convergence in Hilbert c* module
For a monotone sequence of projections in Hilbert c* module, do we have similar conclusion as in Hilbert space (such as strong convergence)? If not, what could be said in that situation?
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Quantum Field theory - integral notation
I have a problem with understanding how the resolution of the identity of an operator is presented in some literature for physicists.
I'm a student of mathematics, and I understand the notion of a ...
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Is the space $S'(\mathbb{N})$ of slowly increasing sequences the projective limit of Hilbert sequence spaces?
Let $S(\mathbb{N})$ be the space of rapidly decreasing sequences and $S'(\mathbb{N})$ its topological dual, the space of sequences bounded by a polynomial.
For $m\in \mathbb{Z}$, we also define ...
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66 views
Existence of a complementary closed subspace extending a given subspace
Let $H$ be an infinite dimensional (separable) Hilbert space (or any infinite dimensional Banach space in which every closed subspace has a complementary subspace). Suppose that $X$ and $Y$ are closed ...
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159 views
Equivalence of Gaussian measures
Let $H$ be a separable Hilbert space and $N(0, C)$ and $N(0, D)$ be Gaussian measures on it. Further, for each $v \in H$, define $R_v = \frac{\left\langle v,Cv \right\rangle}{\left\langle v,Dv ...
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377 views
What is the idea behind interpolation spaces?
I am working through a text on Numerics for SPDEs and there the concept an interpolation (Hilbert-)space associated to an operator is used. To be specific:
Definition. Let $H$ be an ...
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60 views
Local limit theorem for an infinite dimensional integer lattice
Can someone refer me to a local limit theorem for the sum ${\bf S} = \sum_{i=1}^n{\bf X}_i$ of a sequence of independent and identically distributed $d$-dimensional random variables $\{{\bf ...
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78 views
Schur's lemma for antiunitary operators on complex Hilbert spaces
Suppose to have a linear irreducible unitary representation $\rho:G\rightarrow U(H)$ on a complex Hilbert space $H$ with $G$ a generic group. Let $A$ be an $\textit{anti}$-linear operator such that
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Proving that a space is Hilbert
Let $H=H_0^1(0,\ell)\times H_*^1(0,\ell)\times H_*^1(0,\ell)$ be equipped with the norms
\begin{align*}
...
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99 views
A Hilbert-space completion of a Hilbert $ C^{*} $-module over a separable $ C^{*} $-algebra
Let $ B $ be a separable $ C^{*} $-algebra and $ \mathcal{E} $ a Hilbert $ B $-module. We know that $ B $ has a faithful state $ \phi $. Using $ \phi $, we can construct a $ \mathbb{C} $-valued ...
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Two isomorphic Gelfand triplets, is there a problem?
For $j=1, 2$, let $V_j \subset H_j$ be a dense and continuous embedding with $V_j$ a Banach space and $H_j$ a Hilbert space. Identify $H_1 = H_1^*$ and $H_2 = H_2^*$ (using Riesz representation) so ...
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1answer
95 views
ODE system has zero as the only solution?
Let $V \subset H$ be a continuous, compact and dense embedding with $V$ and $H$ Hilbert spaces.
Let $\beta_j:[0,T] \to \mathbb{R}$ be functions for each $j$, and let $v_j$ be a basis of $V_0$.
...
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1answer
104 views
Eigenvalues and Compact Resolvent
For $A$ an unbounded (densely defined) operator on a separable Hilbert space, what conditions on its eigenvalues will show that, for $\lambda \notin $spec$(A)$, we have that $(A-\lambda)^{-1}$ is a ...
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99 views
Bounded operators with infinite matrix representations
I asked this question on StackExchange originally, but I'm giving it a go here as well.
Suppose that $A$ is a unital $C^*$-algebra, $\varphi\colon A\to B(H)$ is a unital, completely positive map and ...
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RKHS norm and posterior of Gaussian process
In Srinivas et al (2010) [appendix B], the authors claim the following "easy to see" property relating the norm of a function in a RKHS induced by a kernel $k(\cdot,\cdot)$, and its norm in the RKHS ...
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Examples for Markov generators with pure point spectrum
I'm looking at symmetric diffusion Markov generators $L$ with pure point spectrum, i.e. infinitesimal generators of symmetric diffusion Markov semigroups, which are defined on $L^2(\mu)$ where $\mu$ ...
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Hilbert triples
I apologize if this is a trivial matter for the analysts out there, but I looked this up in a number of books (e.g., H. Brezis, J. Wloka) and could not find a satisfying discussion, one that would ...
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A linear operator equation (PDE) with non-monotone term
I'm interested in the existence and/or uniqueness to the following problem. Let $V$ and $H$ be Hilbert spaces and $V \subset H \subset V^*$ form a Gelfand triple.
There is a linear operator $L:{D}(L) ...
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115 views
Fractional Brownian motion via Hilbert space
The Brownian motion has the following (Levy-Ciesielski?) construction via Hilbert space isomorphisms:
Let $\{ Z_i \}_{i \in \mathbb{Z}}$ be i.i.d. $N(0,1)$ random variables defined on $(\Omega, ...
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113 views
Invertible unbounded linear maps defined on a Hilbert space
It is well-known that, assuming the axiom of choice, there are unbounded linear maps defined not only on a dense subset but on all of Hilbert space. Is it possible that such a map is invertible?
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186 views
Positive definite quadratic forms on Banach spaces
This is a question about characterizing Hilbert spaces in terms of quadratic forms. Let $X$ be a real Banach space and $E$ a bounded quadratic form on it, it is called positive definite if ...
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216 views
How to Prove the Semi-parametric Representer Theorem
I have also posted this question on MathSE, so if you think it mustn't be here, please let me know, or just delete it. I was thinking that some people here are more appropriate to answer this ...
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Measurability of a map that takes a functional to its composition with a linear operator
Let $(X,\Sigma_X)$ be a measurable space such that $\Sigma_X$ is countably generated.
Let $B_b(X)$ be the Banach space of all bounded $\Sigma_X$-measurable functions $X\to\mathbb{R}$ equipped with ...
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38 views
Speed of convergence of vector expansions in non orthogonal basis
Suppose we have a finite-norm vector $X$ in a Hilbert space, and we want to construct its expansion in a certain (infinite) basis $V_k$, $X=\sum_k a_k V_k$. If the basis is orthonormal, then we know ...
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63 views
Integral representation of joint projection valued measures
Given two positive $\sigma$-finite measures $\mu_{1/2}$ on the spaces $X_{1/2}$ one can define the product measure $\mu_1\otimes\mu_2$ on the product space $X_1\times X_2$. It can be proved that the ...
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Determining the exact form of a projection in a Hilbert space
Let $$\Omega = \left\{f(x) \in \mathcal{L}^2[0,T]: \frac{1}{T}\int_0^Tf(x)dx = \mu,~ a \le f(x) \le b,~\forall x \in [0,T]\right\},$$
where $\mathcal{L}^2[0,T]$ is the set of Lebesgue ...
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For which $f \in L^2([0,1])$ is $f^\perp \cap C^\infty$ dense in $f^\perp$?
Given $f \in L^2([0,1])$, $f \neq 0$, we can consider the orthogonal complement $f^\perp$ . The smooth functions $C^\infty([0,1])$ are dense in $L^2([0,1])$. Is the intersection $f^\perp \cap ...
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is 1/max(i,j) a bounded matrix on hilbert spaces?
I would like to know if the infinite matrix $[\frac{1}{\max(i,j)}]_{i,j\geq 1}$ represents a bounded operator on $\ell^2(\mathbb{N}^\star)$.
It would be sufficient to know if the Lehmer matrix ...
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Hilbert Schmidt Operators and the Conditional Expectation Operator
Consider the function $\text{E}_W: L_2(\mathbb{R},P_X) \mapsto L_2(\mathbb{R},P_W)$ where $P_X$ and $P_W$ are two different probability measures. They are related in such a way that if $f_X$, $f_W$ ...
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171 views
what characterizes a characteristic function of a probability measure in separable Hilbert spaces?
As we all know on real line $\mathbb{R}$, the following is valid
A $\mathbb{C}$-valued function $\varphi$ is a characteristic function of a probability measure on $\mathbb{R}$ if and only if ...
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Nuclear vs Integral operators on Hilbert spaces
Consider the Hilbert space $L_2=L_2[0,1]$. Is it true that for each nuclear (trace-class) operator on $L_2$ there exists a function $K\in L_1(L_2)$ such that
$$Tf = \int\limits_0^1 K(s) f(s) ...
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164 views
Karhunen-Loeve expansion for discrete-time process
Is there a Karhunen-Loeve theorem for discrete-time process?
For example, let $\left\{X_i\right\}$ be a sequence of independent random variable which are uniformly distributed on the set $\{-1,1\}$. ...
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362 views
Spectral decomposition of compact operators
Let T be a compact operator on an infinite dimensional hilbert space H. I am proving the theorem which says that $Tx=\sum_{n=1}^{\infty}{\lambda}_{n}\langle x,x_{n}\rangle y_{n}$ where ($x_{n}$) is an ...
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Closure of partial differential operators on $L^2(\Omega)$
Let $\Omega\subset\mathbb{R}^2$ open set. Consider an $\textit{uniformly elliptic}$ second order differential operator on $L^2(\Omega,\mathbb{C})$
$$
H=\sum_{|\alpha|\le 2}C_\alpha\partial^\alpha
$$
...
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91 views
Does positivity preserve compactness? [closed]
Suppose $A$ and $B$ are operators on a (separable) Hilbert space $H$ and $A \leq B$. Is it true that if $B$ is compact then $A$ is compact too? If not, could you please show a counterexample?
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Besicovitch Almost Periodic Functions a subspace of what?
The common example of a nonseparable Hilbert space comes from the collection of Besicovitch almost periodic function spaces. Starting with $L^p_{\text{loc}}(\mathbb{R})$ we look at those elements ...
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Characterisation of compact operators
It is known (and easy) to prove that if $T: H\longrightarrow H $ is compact, where $H$ is a Hilbert space, then for any orthonormal basis $ e_n $ we have $||Te_n||\longrightarrow 0$.
My question is ...
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A Theorem by Von Neumann, which pertains a product of two Hilbert Spaces
I'm writing my thesis on the EPR paradox (I want to continue my master degree in physics) but I'm having an unusual problem. One passage from the book I'm following at the moment justifies one ...
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Is there a nice “minimum” of two symmetric operators?
Let $A$ and $B$ be two bounded symmetric positive operators in Hilbert space, such that $A-B$ is trace class. If needed, $A$ and $B$ may be assumed reasonably "small", let's say, Hilbert-Schmidt.
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Exterior powers and singular values on Hilbert spaces
I am currently writing an article relating to multiplicative ergodic theorems for cocycles of bounded operators acting on a Hilbert space, and in parts of the argument it is necessary for me to refer ...
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A question about tiling Hilbert Space
Let H be an infinite dimensional and separable Hilbert Space. Let e be a positive real number-which can be arbitrarily small. Does there exist a denumerably infinite set S of pairwise disjoint and ...
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1answer
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Does spectral theory assume separability
On an infinite dimensional space, the spectral theorem for compact normal operators says that the eigenvectors form an orthonormal basis which, from wikipedia, is equivalent to the space being ...
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Special form of unbounded operators on $L_2(\mathbb{R}_+, \mathcal{H})$
I have the following problem;
Fix a Hilbert space $\mathcal{H}$. Let $S \colon \mathrm{Dom}S \subset L_2(\mathbb{R}_+, \mathcal{H}) \rightarrow L_2(\mathbb{R}_+, \mathcal{H}) $ be a closed densely ...
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572 views
Convex hulls of compact sets
Let $A$ be a compact set in a separable Hilbert space $H$, and let $\bar A$ denote its convex hull. Is $\bar A$ compact?
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Isometric embeddings of metric spaces in Hilbert spaces
There are plenty of isometric embeddings of metric spaces in Banach spaces. Nevertheless, I have been unable to find any significant result on isometric embeddings into Hilbert spaces. My question is: ...
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Cardinality of the set of Boolean subalgbras of the lattice of projections on a Hilbert space.
A simple question I've managed to gey myself quite confused about.
Given a Hilbert space H, what do we know about the cardinality of
(a) the set P(H) of projection operators onto H (equivalently, ...
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compact-open topology on $B(H)$
In topology, it is common to use the compact-open topology on the set of continuous maps between two given topological spaces.
Let now $H$ be a Hilbert space and $B(H)$ the set of continuous linear ...
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status of Invariant subspace problem on Krein Space
What is the status of Invariant subspace problem on Krein Space? What sort of developments have taken place in this area.
