The hilbert-spaces tag has no wiki summary.
0
votes
0answers
39 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
74 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
132 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
176 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
192 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
175 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
138 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
252 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
82 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
92 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
119 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
328 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
111 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
266 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
64 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
30 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
$$H^2(M) = \{f \in L^2(M)\ |\ \bar{\partial}_{b}f = 0\ ...
0
votes
0answers
62 views
Interpret some coefficients in algebras
Let $A$ be a real vector space equipped with a scalar product $\langle \,,\,\rangle$, and assume moreover that a multiplication is define on it so that becomes an algebra (e.g. polynomials with the ...
5
votes
1answer
123 views
An extreme point of the ball of the space of compact operators
It is very easy to see that the unit ball of $c_0$ has no extreme points. I was trying to spot any extreme points in the unit ball of the space of compact operators on a Hilbert space (a ...
1
vote
0answers
51 views
“correlation” between two subspaces
It is frequently important to determine the coherence of a matrix by finding the maximum pairwise correlation between all its column vectors. Similarly, when working with a union of subspaces of a ...
2
votes
1answer
107 views
Iterated projections in Hilbert spaces
Let $E$ be an Hilbert space and $F, G$ two subspaces such that $F \cap G =\{0\}$. Let $(x_n)$ be the sequence of iterated orthogonal projections: $x_0 \in F$, $x_1$ is the orthogonal projection of ...
2
votes
0answers
79 views
Terminology and reference question
I am working on a problem involving bilinear forms over complex Hilbert spaces, and in my case it is not natural to make the forms sesquilinear, i.e., $a(u,v)$ is linear in both complex arguments. ...
0
votes
1answer
117 views
find a weak solution in an intersection of Sobolev spaces
In
using-lax-milgram-to-find-a-weak-solution-in-an-intersection-of-sobolev-spaces
the weak solution for
$$
-\Delta^2 u = f \in L^2(U)\\ \\
u|_{\partial U}=\Delta u|_{\partial U} = 0
$$
was ...
0
votes
0answers
84 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
139 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
107 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
185 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
186 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
96 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
136 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
95 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
135 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
152 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
102 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
187 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
169 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
544 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
231 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
488 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
105 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)$$
we wish to recover the coefficients $a_{n}$ . the only thing i though would work is to try and come up with a function ...
6
votes
1answer
217 views
A doubt about the parts of the spectrum of tensor products
Let $\mathcal{H}$ be any complex Hilbert space of infinite dimensional. By an operator $T$ I mean a linear bounded transformation from $\mathcal{H}$ into $\mathcal{H}$, i.e, ...
4
votes
3answers
266 views
ordered exponential of unbounded operators
Let $H$ be a Hilbert space,
and let $A_t$ be a family of unbounded positive (self-adjoint) operators on $H$ parametrized by $\mathbb t\in R_{\ge 0}$. Consider the ordinary differential equation
$$
...
5
votes
2answers
264 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
407 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 $C \subset \mathbb{P}^3$ ...
0
votes
1answer
118 views
Rational equivalence and Hilbert flag scheme
Given a smooth surface $X \subset \mathbb{P}^3$ if we have two curves $C_1, C_2$ that are rationally equivalent is it true that both $(C_1,X)$ and $(C_2,X)$ will be in the same irreducible component ...
0
votes
1answer
358 views
A doubt about tensor product on Hilbert Spaces
An operator is a bounded (i.e., continuous) linear transformation between Hilbert spaces. Let $\mathcal{B}[\mathcal{H}]$ be the set of all operators in the Hilbert space $\mathcal{H}$.
Let ...
2
votes
1answer
130 views
Partial isometries making families of linearly independent vectors orthogonal
Suppose I have a family of $n$ linearly-independent elements $v_i$ of the Hilbert space $\mathbb{C}^m$, which are not necessarily orthogonal. Can I always find a partial isometry $f: \mathbb{C} ^m \to ...
