All Questions
467 questions
4
votes
0
answers
404
views
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 $f_X$, $f_W$ ...
- 289
4
votes
0
answers
277
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 ...
- 6,206
4
votes
0
answers
112
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.
- 2,106
3
votes
3
answers
2k
views
Determining if a set is a Basis for l^2
For each $ n\ge 1$ Define the vectors $e_n = (e_{nk})$ where $ k\ge 1$ and $ e_{nk} = \frac{1}{k^n}$
Is this set a basis for $l^2$?
Thanks,
- 4,143
3
votes
4
answers
1k
views
Is there a use for a Hilbert space that uses a different norm than the one induced by the inner product?
$l_1$ minimization / compressed sensing comes to mind. Does anyone have any concrete examples? Or is such a construct completely useless?
- 49
3
votes
1
answer
555
views
Trace norm of operators obtained by restricting the matrix of a trace class operator
Suppose $H$ is a Hilbert space with orthonormal basis $\{e_i\}_{i\in \mathbb N}$. To every operator $T$, we associate a infinite matrix $[T_{ij}]$, where $T_{ij}=\left<Te_j,e_i\right>$. We know ...
- 243
3
votes
3
answers
319
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 (...
- 1,361
3
votes
2
answers
570
views
Matrices Representing Bounded Operators and Absolute Values
Let $A=(a_{ij})_{i,j=1}^{\infty}$ be an infinite matrix of complex numbers. For every positive integer $n$, we shall denote with $A_n$ the $n \times n$ matrix $A_n=(a_{i,j})_{i,j=1}^{n}$, and if $x \...
- 640
3
votes
1
answer
285
views
Closable unbounded operators and Banach space adjoints
For an unbounded operator $T:\mathcal{H}_1 \to \mathcal{H}_2$, if its adjoint $T^*$ is densely defined, then we know that $T$ is closable. What happens if we replace $\mathcal{H}_1$ or $\mathcal{H}_2$ ...
- 969
3
votes
1
answer
332
views
What is the consistency strength of non-existence of outer automorphisms of Calkin algebra?
The Calkin algebra $C(H)$ is the quotient of $B(H)$, the ring of bounded linear operators on a separable infinite-dimensional Hilbert space $H$, by the ideal $K(H)$ of compact operators.
In 1977, ...
3
votes
2
answers
816
views
How to determine if a given set of polynomials has dense linear span in $L^2([0,1])$?
Consider the following set of polynomials: $S := \{x^{2m}: m \geq 0\}$. Is there a non-zero element $f \in L^2([0,1])$ such that $\int_0^1 fx^{2m} = 0$ for each $m \geq 0$? Note that the answer is ...
- 5,339
3
votes
1
answer
598
views
is a non-invertible operator a boundary point of the group of invertible operators?
Good evening,
I have a question concerning non-invertible operators.
Let $H$ be a Hilbert space and $T$ a non-invertible bounded operator on $H.$ Is it true that $T$ is the limit of some sequence ...
- 758
3
votes
1
answer
426
views
Explicit example $f_k \to f$ converging strongly in $L^6(R^3)$, but only weakly in $H^1(R^3)$
Can we find an explicit example of a sequence of functions $f_k \in H^1({\mathbf R}^3)$ such that, $f_k \rightharpoonup f$ weakly converges in $H^1({\mathbf R}^3)$ and $f_k \to f$ strongly converges ...
- 543
3
votes
1
answer
214
views
Non-point spectrum for diagonalisable self-adjoint unbounded operator
Given a (separable) Hilbert space H and an unbounded densely defined linear operator $T:{\cal D}(T) \to $H such that ${\cal D}$ is diagonalizable (it means $\exists$ an O.N.B. of H such that all basis ...
- 969
3
votes
1
answer
1k
views
If $H$ is a separable Hilbert space, is its dual dense in $L^2(H)$?
Let $H$ be an infinite-dimensional, separable Hilbert space, and let $\gamma$ be a Radon probability measure on $H$ with mean zero and covariance operator the identity $I$.
Let $H^*$ denote the space ...
- 8,512
3
votes
1
answer
1k
views
Closure of polynomials of a function in $L^2$
Suppose $f \colon I \to \mathbb{R}$ is a function in, say, $L^\infty$, and $I \subset \mathbb{R}$ is a bounded interval. We may assume further regularity on $f$, such as Lipschitz continuity or strict ...
- 648
3
votes
1
answer
157
views
Operator in the commutant which is small on a given vector
Suppose $x$ is a non-zero vector in a Banach space, and $T$ is a fixed operator. Is the following true:
For any $\varepsilon, \delta$, there exists $S$ in the commutant of $T$ such that $1\leq\|S\|<...
- 1,361
3
votes
1
answer
170
views
A specific problem on : Can bounding the Sobolev norm, bound a higher derivative?
Let $f \in H^k(\mathbb{R}^m)$, $k>\frac{m}{2}$. Given any $f$, such that $\|f\|_{H^k(\mathbb{R}^m)}<K$ , and any $\phi \in C^{\infty}(\mathbb{R}^m)\cap H^k(\mathbb{R}^m)$, such that $\|\phi\|_{...
- 698
3
votes
1
answer
261
views
Can the $L^{\infty}\to L^{\infty}$ norm be bounded by the trace norm?
Let $k\in C(\mathbb{R}^2; \mathbb{R})$ be a continuous function. Suppose that the operator $K\colon L^2(\mathbb{R}) \rightarrow L^2(\mathbb{R})$ defined by the formula
$$(Kf)(x)=\int_{\mathbb{R}} k(x,...
- 128k
3
votes
1
answer
159
views
Tight L2 bound on moments approximation and reference
Consider $f\in L^2(I)$, where $I$ is the unit interval and $L^2$ is w.r.t. Lebesgue measure, and consider an approximation of $f$ denoted by $\tilde{f}\in L^2$.
The error in approximated the moments ...
- 3,574
3
votes
2
answers
6k
views
Need help understanding Riesz representation theorem for Reproducing Kernel Hilbert Spaces
I'd like some help understanding any of the following proofs of Riesz representation theorem -- whichever is simpler -- or in fact any proof of the theorem.
Proof 1: http://nfist.pt/~edgarc/wiki/...
- 661
3
votes
1
answer
229
views
Symmetric diagonalizable operators and self-adjointness
Given a densely defined symmetric operator $L$ on a Hilbert space $H$, which is also assumed to be diagonalizable, will there always exist a unique extension of $L$ to a self-adjoint operator?
3
votes
1
answer
475
views
Extension of a bounded linear functional
Let $\mathcal{H}$ be a finite-dimensional Hilbert space and $\mathcal{A}\subseteq\mathcal{B(\mathcal{H})}$ be an operator system. Suppose $T_n$ is a collection of all $n$-by-$n$ matrices equipped with ...
- 231
3
votes
1
answer
148
views
Is this sum of nonexpansive maps itself nonexpansive?
For Hilbert spaces $\mathcal{H}_X$, $\mathcal{H}_Y$ and $\mathcal{K}$, consider a linear map $f\colon \mathcal{K} \oplus \mathcal{H}_X \to \mathcal{K} \oplus \mathcal{H}_Y$ that is given as a matrix
$...
3
votes
1
answer
348
views
Completely Positive Maps and their dual in Separable Hilbert Space
Consider a separable Hilbert space $\mathcal{H}$, we can first define $T(\mathcal{H})$ the trace class of $\mathcal{H}$, then $D(\mathcal{H})$ to denote the set of positive operator with trace less ...
- 1,503
3
votes
1
answer
266
views
Convergence of nuclear operators
Let $H$ be a separable infinite-dimensional real Hilbert space. We consider operators in $H.$
Nuclear norm of a nuclear operator is the sum of its singular values.
A nuclear, positive and self-...
3
votes
1
answer
397
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, \...
- 273
3
votes
1
answer
174
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 ...
- 3,115
3
votes
1
answer
1k
views
Self-adjoint bounded operator, resolution of the identity, def. of the diagonal
Let $A$ be a self adjoint bounded linear operator with a continuous spectrum
$\sigma(A)=[a,b]$ which acts on a separable Hilbert space. Let
$E_\lambda$ be its resolution of the identity.
For ...
3
votes
1
answer
209
views
Must solutions to the time-independent Schrodinger equation that have discrete or negative eigenvalues be square-integrable?
This is a very straightforward question (although the answer may not be!) about a ubiquitous textbook physics situation. I assume that the answer is probably well-known, but I asked it on both Math ...
- 1,311
3
votes
2
answers
135
views
Unicellular compact operators
An operator $T$ on a separable Hilbert space $H$ is called unicellular if any two closed invariant subspaces $M$ and $N$ are comparable; that is either $M\subseteq N$ or $N\subseteq M$. There are many ...
- 1,361
3
votes
1
answer
261
views
norm estimates for Schatten class
Let $C
_p$ be the Schatten-p-classes on a separable Hilbert spaces, $p\ge 1$.
Let ${\rm Tr}$ be the standard trace.
Let $y\in C_p$ be a self-adjoint operator (or even a positive operator) and let $...
- 617
3
votes
1
answer
176
views
Dense subspace of $\operatorname{Ind}_{H_1 \times H_2}^{G_1 \times G_2} \chi$
Let $H = H_1 \times H_2$ be a closed subgroup of a second-countable locally compact Hausdorff group $G = G_1 \times G_2$, with $H_i \leq G_i$. Let $\chi = \chi_1 \otimes \chi_2$ be a unitary ...
- 6,180
3
votes
1
answer
90
views
Measurability of specific function
Let $I\subset\mathbb{R}$ denote an open and bounded interval of the real line, $H_0^1(I)$ all quadratic integrable Sobolev functions and $C(\bar{I})$ all continuous functions on said interval.
Since ...
- 65
3
votes
2
answers
471
views
inner product on matrix spaces of multivariate polynomials?
Let $H_{n,d}=\mathbb{R}_d[x_1,..,x_n]$ be the space of $n$-variate homogeneous degree $d$ polynomials, $D=D^\top\in \mathbb{N}^{m\times m}$ a symmetric $m\times m$ matrix. Consider the space $P_D$ of ...
- 14k
3
votes
1
answer
180
views
Are the paths of the Brownian motion contained in a suitable RKHS?
Let $H_B$ be the reproducing kernel Hilbert space (RKHS) of the Brownian Motion $(B_t)$ on $[0,1]$. It is well known that with probability 1 the paths of $(B_t)$ are not contained in $H_B$.
But is ...
- 31
3
votes
1
answer
140
views
Infinite-dimensional analogue of "positive-negative splitting implies non-degeneracy"
(This question is related to Splitting a space into positive and negative parts but different.)
Given a finite-dimensional vector space $V$ over $\mathbb{R}$, what I call a "positive-negative ...
- 1,804
3
votes
1
answer
497
views
Hilbert-Schmidt integral operator with missing eigenfunctions
I'm having some issues with the spectral decomposition of the integral operator
\begin{equation}
(Af)(x)=\int_0^1|x-y|f(y)dy,\text{ with $f\in L^2[0,1]$}.
\end{equation}
Since
\begin{equation}
...
- 33
3
votes
1
answer
170
views
inequality involving tuple of operators on Hilbert spaces
Let $E$ be a complex Hilbert space.
Let $(A_1,...,A_n) \in \mathcal{L}(E)^n$, could you please help me to show that
$$\displaystyle\sup_{\|x\|=1}\bigg(\displaystyle\sum_{i=1}^n\| A_i^*x\|^2\bigg)\...
- 1,154
3
votes
1
answer
97
views
Hypercontractions and automorphisms of the unit disc
Recall that an bounded operator $T$ on a Hilbert space $\mathcal H$ is said to be $n$-hypercontraction for $n\in\mathbb N$ if
$$ I- {n \choose 1} T^*T + {n \choose 2} {T^*}^2T^2-\cdots+ (-1)^{n}{n \...
- 207
3
votes
1
answer
275
views
Sum of two parts of a continuous stochastic process
Let $X$ be a centered continuous stochastic process which is square integrable on $[0,2]\times \Omega$ and the basis of $L^2(0,2)$ is $\{e_i\}$. By using Karhunen-Leove Theorem one can write for all $...
- 213
3
votes
1
answer
870
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\}$. ...
user24451
3
votes
1
answer
3k
views
Approximate point spectrum
I have a question concerning the relation between the approximate point spectrum and the spectrum of an operator.
Let $T$ be a bounded linear operator of a complex Hilbert space $H$. The approximate ...
- 758
3
votes
1
answer
588
views
orthonormal basis of eigenvectors for laplacian on a concave polygon
I am interested in the Laplace operator $\Delta$ on a concave polygon.
When the polygon is convex, it is known that $\Delta: H^2(\Omega) \rightarrow L^2(\Omega)$
is boundedly invertible. In addition, ...
- 31
3
votes
1
answer
914
views
Range of a Certain Linear Operator
Consider the following hermitian form on the sobolev space H^1(I), of an interval I:
g(u,v):= \int_I (du/dt dv/dt - \rho(t) u v)dt, where \rho is a nice bounded function on I.
Riesz representation ...
- 31
3
votes
0
answers
95
views
Commutator of $A\otimes I$ and $I \otimes B$ vanishes?
Consider two Hilbert spaces $H_1$ and $H_2$, and $A$, $B$ unbounded operators on $H_1$, $H_2$ respectively. $(A \otimes I)$ is classically defined as the closure of the operator defined on the set of ...
- 31
3
votes
0
answers
104
views
Comparing unitaries which are perturbatively close
Let $\mathcal{H}$ be a Hilbert space and let $H_0$ and $H_1$ be two Hermitian operators on $\mathcal{H}$. Thinking of $H_1$ as a perturbation of $H_0$, the Duhamel formula allows us to write $e^{-...
- 452
3
votes
0
answers
198
views
On a paper of von Neumann
Let $H$ be a Hilbert space and $T: H \to H$ be a contraction. In Eine Spektraltheorie für allgemeine Operatoren eines unitären Raumes, von Neumann proved the inequality
$$
\lVert p(T)\rVert \leq \sup \...
- 31
3
votes
0
answers
122
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 ...
- 561
3
votes
0
answers
390
views
How to prove the polar decomposition of unbounded operators?
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)} $ ...
- 1,219