The tag has no wiki summary.

learn more… | top users | synonyms

1
vote
0answers
86 views

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 ...
2
votes
1answer
66 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 $\{-1,1\}$. ...
0
votes
2answers
227 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 ...
0
votes
0answers
51 views

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 $L^2(\Omega,\mathbb{C})$ $$ H=\sum_{|\alpha|\le 2}C_\alpha\partial^\alpha $$ ...
0
votes
2answers
83 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?
4
votes
1answer
145 views

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 ...
1
vote
2answers
192 views

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 ...
4
votes
2answers
197 views

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 ...
6
votes
1answer
184 views

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. ...
3
votes
0answers
139 views

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 ...
9
votes
1answer
260 views

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 ...
0
votes
1answer
88 views

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 ...
1
vote
1answer
94 views

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 ...
2
votes
0answers
132 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?
0
votes
2answers
338 views

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: ...
3
votes
1answer
113 views

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, ...
12
votes
2answers
279 views

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 ...
3
votes
0answers
65 views

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.
1
vote
0answers
31 views

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\ ...
0
votes
0answers
63 views

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 ...
5
votes
1answer
128 views

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 ...
1
vote
0answers
56 views

“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 ...
2
votes
1answer
109 views

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 ...
2
votes
0answers
80 views

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. ...
0
votes
1answer
122 views

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 ...
0
votes
0answers
88 views

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 ...
7
votes
2answers
141 views

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 ...
2
votes
1answer
114 views

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 ...
1
vote
2answers
196 views

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 ...
7
votes
1answer
189 views

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 ...
0
votes
1answer
99 views

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, ...
3
votes
0answers
138 views

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, ...
2
votes
1answer
100 views

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 ...
1
vote
1answer
121 views

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 ...
0
votes
1answer
141 views

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 ...
3
votes
1answer
118 views

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 ...
3
votes
1answer
156 views

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. ...
0
votes
0answers
104 views

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 > ...
1
vote
1answer
191 views

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, ...
2
votes
1answer
175 views

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 ...
0
votes
2answers
580 views

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 ...
2
votes
1answer
235 views

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 ...
4
votes
1answer
493 views

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 ...
1
vote
0answers
109 views

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 ...
6
votes
1answer
219 views

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, ...
4
votes
3answers
276 views

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 $$ ...
5
votes
2answers
268 views

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 ...
2
votes
2answers
414 views

Can one derive Wigner’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 ...
1
vote
1answer
138 views

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 ...
2
votes
1answer
262 views

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 $C \subset \mathbb{P}^3$ ...