The hilbert-spaces tag has no wiki summary.

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### Proving that a positive operator has a unique square root [migrated]

Rudin's functional analysis page 331 theorem 12.33) proves this. He proves uniqueness by 'going back' to the general algebra setting. I was just wondering whether there is a more direct way of doing ...

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### 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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### 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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### 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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### 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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### 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.

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### Is the Szego projection on a codim-$k$ CR manifold an integral operator?

The Szego projection on a CR manifold $M$ is defined to be the orthogonal projection from $L^2(M)$ to the closed subspace $H^2(M)$ where
$$H^2(M) = \{f \in L^2(M)\ |\ \bar{\partial}_{b}f = 0\ ...

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### Interpret some coefficients in algebras

Let $A$ be a real vector space equipped with a scalar product $\langle \,,\,\rangle$, and assume moreover that a multiplication is define on it so that becomes an algebra (e.g. polynomials with the ...

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### An extreme point of the ball of the space of compact operators

It is very easy to see that the unit ball of $c_0$ has no extreme points. I was trying to spot any extreme points in the unit ball of the space of compact operators on a Hilbert space (a ...

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### “correlation” between two subspaces

It is frequently important to determine the coherence of a matrix by finding the maximum pairwise correlation between all its column vectors. Similarly, when working with a union of subspaces of a ...

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### Iterated projections in Hilbert spaces

Let $E$ be an Hilbert space and $F, G$ two subspaces such that $F \cap G =\{0\}$. Let $(x_n)$ be the sequence of iterated orthogonal projections: $x_0 \in F$, $x_1$ is the orthogonal projection of ...

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### Terminology and reference question

I am working on a problem involving bilinear forms over complex Hilbert spaces, and in my case it is not natural to make the forms sesquilinear, i.e., $a(u,v)$ is linear in both complex arguments. ...

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### find a weak solution in an intersection of Sobolev spaces

In
using-lax-milgram-to-find-a-weak-solution-in-an-intersection-of-sobolev-spaces
the weak solution for
$$
-\Delta^2 u = f \in L^2(U)\\ \\
u|_{\partial U}=\Delta u|_{\partial U} = 0
$$
was ...

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### Operator Adjoints and Non-Symmetric Inner Products

Let $V$ be a finite dimensional vector space (over $C$ if that makes a difference), and let $T$ be a linear operator on $V$. Now if $(\cdot,\cdot)$ is an inner product on $V$, then it is well-known ...

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### why is this a sufficient condition for a domain to be a core of an unbounded operator?

Let $\alpha:\mathbb R\to U(H)$ be a strongly continuous action of the reals on some Hilbert space, and let $A=-i\frac d{dt}\alpha(t)|_{t=0}$ be its infinitesimal generator, so that ...

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### Self-adjointness of a perturbed quantum mechanical Hamiltonian specified in an infinite matrix form

Consider an operator $H$ on the Hilbert space $\ell_2$ given as an infinite matrix with two pieces, one diagonal and one arbitrary:
$H_{ij}=E_i\delta_{ij}+V_{ij}$. This has a physical meaning in ...

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### Finite dimensional approximations of operators on Hilbert spaces

Let $e_1,e_2,\dots$ be a Schauder basis for a Hilbert space $(V , \langle \cdot , \cdot \rangle)$. Let $A:V \to V$ be an operator. Finally, let $V_n = {\rm span}( e_1, \dots, e_n)$. Let $i_n : V_n ...

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### Product of commuting nonnegative operators

Let $V$ be a real vector space with an inner product and $A,B : V \to V$ linear maps which are self-adjoint nonnegative-definite, i.e. $\langle Ax,y \rangle = \langle x,Ay \rangle$ and $\langle Ax,x ...

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### Reference for compact operators in quaternionic Hilbert spaces

Have they been studied? In particular, what is the analogue of the Schmidt theorem for compact operators in Hilbert spaces?
Helemskii A. Ya., Lectures and Exercises on Functional Analysis, Ch. 3, ...

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### Deleting “weak homeomorphism” in a Hilbert space

It is well-known that there exists a homeomorphism $h$
from an infinite-dimensional Hilbert space $H$ to $H\setminus\{0\}$.
Does there exist a "weak homeomorphism" $g:H \to H\setminus\{0\}$,
that is, ...

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### RKHSs containing constant functions

Suppose $H$ is the reproducing kernel Hilbert space on a space $X$ with reproducing kernel $K$. If, say, $K - c$ is a positive definite kernel for some $c>0$ then $H$ contains the constant ...

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### Special kind of operators

Consider an operator $A: H \longrightarrow X$ ($H$ is a Hilbert space and $X$ is a Banach space) that has a representation
$$ A = \sum_{j=0}^\infty a_j \langle \cdot, e_j\rangle \cdot x_j,$$
where ...

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### Can an accumulation point be an eigenvalue?

For an discrete (separable) infinite-dimensional Hilbert Space with a compact operator, 0 is always an accumulation point (https://www.math.ucdavis.edu/~hunter/book/ch9.pdf).
Does this mean its part ...

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### A space parameterizing the choices of orthonormal bases for a Hilbert space

Let $\mathcal{H}$ be an infinite dimensional separable (complex) Hilbert space. What is a natural space which parameterizes the choices of orthonormal bases for $\mathcal{H}$?
It seems like one ...

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### On the self-adjoint part of a quasinilpotent operator

Disclaimer: this is not research-level, but I've read some non research-level questions/answers on quasinilpotent operators here, some of them involving renowned users. So I thought I'd give it a try. ...

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### separable spaces-QM vs. functional analysis

In Nouredine Zettili's QM book, Hilbert space $H$ is said to be separable when: There exists a Cauchy sequence $\psi_n \in H$ ($n=1,2,\ldots)$ such that for every $\psi$ of $H$ and $\varepsilon > ...

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### Do kernels provide a basis for a RKHS?

Let $H$ be a Reproducing Kernel Hilbert Space with elements $f:X\rightarrow \mathbb{C}$, with kernel $K(x, y)$. My question is whether, for some choice of $x_i\in X$, it is the case that $u_i:=K(x_i, ...

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### Coercive Symmetric Bilinear form on a Hilbert space

I need to show one of the two following equivalent results. If true, it must be a simple proof but I do not seem to be able to make it work. Thank you in advance.
1) Consider a continuous symmetric ...

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### When are two operators simultaneously diagonalisable?

I am reading a paper and they have diagonalised both operators in an equation, on a separable Hilbert space, with respect to the same basis. My question is, when can two operators be simultaneously ...

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### Decomposing bilinear forms in Hilbert spaces

You are given a complex Hilbert space $H$ with two equivalent Hilbert space structures $<,>$ and $<,>'$. Define $<,>''=<,> + <,>'$ to be the sum of our two scalar ...

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### What sort of manifold is PU(H)?

The space underlying the projective unitary group of a separable, infinite-dimensional Hilbert space has a number of topologies, so for the purposes of this question, pick you favourite and answer for ...

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### an infinite series expansion in terms of the polylogarithm function

we have the complex valued function :
$$f(z)=\sum_{n=0}^{\infty}a_{n}Li_{-n}(z)$$
we wish to recover the coefficients $a_{n}$ . the only thing i though would work is to try and come up with a function ...

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### A doubt about the parts of the spectrum of tensor products

Let $\mathcal{H}$ be any complex Hilbert space of infinite dimensional. By an operator $T$ I mean a linear bounded transformation from $\mathcal{H}$ into $\mathcal{H}$, i.e, ...

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### ordered exponential of unbounded operators

Let $H$ be a Hilbert space,
and let $A_t$ be a family of unbounded positive (self-adjoint) operators on $H$ parametrized by $\mathbb t\in R_{\ge 0}$. Consider the ordinary differential equation
$$
...

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### Limit of simple tensors

I have two questions which are intuitively true.
Let $V$ be a Hilbert space. As usual we can turn $V\otimes V$ or $V\otimes V\otimes V$ into Hilbert spaces by intorducing the natural inner product ...

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### Can one derive Wignerâ€™s theorem in the complex case from the real case?

Wigner's theorem states that every symmetry of complex Hilbert space is either unitary or anti-unitary, up to multiplication by a unit scalar function. Here $f:\mathcal{H} \rightarrow \mathcal{H}$ is ...

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### Weyl asymptotics vs. form perturbations

Consider Hilbert spaces $V$,$H$; a closed quadratic form $a$ with domain $V$; and its associated operator $A$ on $H$. (If necessary, the form can be assumed to be coercive.) For the sake of ...

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### Deformation of space curves to union of lines

Does every irreducible component of a Hilbert scheme of curves in $\mathbb{P}^3$ contain a curve that is a union of lines (not necessarily reduced)? Furthermore, given a curve $C \subset \mathbb{P}^3$ ...