All Questions
Tagged with operator-theory functional-analysis
7,084
questions
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Question about invariant subspaces of C*-algebras.
Let $M$ be a subspace of some Hilbert space $H$ and let $U$ denote a $C*$-algebra contained in $L(H)$. Is it true that closure$[UM]$ is an invariant subspace of $U$?
I believe it is:
Assume $x \in$ ...
0
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0
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31
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Multiplicative Functionals: Intuition
I am trying to get a better intuition on multiplicative functionals over a commutative Banach algebra. The definition to be is clear (simply: unital algebra homomorphisms into $\mathbb{C}$) and I am ...
- 25
1
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0
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41
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Definition of Hilbert-Schmidt integral operator
In the definition of Hilbert Schmidt integral operator, we require the kernel $K(\cdot,\cdot)$ to be defined on $L^2(X\times X)$. I don't quite understand this restriction. Can we allow $K$ defined on ...
- 487
1
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36
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Multiplicity of bilateral shift on a Banach space
Let $\mathbb{X}$ be a Banach space. A bijective linear map $V: \mathbb{X} \to \mathbb{X}$ is said to be a bilateral shift if there is a closed subspace $\mathbb{L}$ of $\mathbb{X}$ such that
$\mathbb{...
- 73
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33
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Whether we can get energy estimate $d\|X\|^2+2\|X\|_{H^1}^2 dt=2(X, B dW) +{\rm Tr}(BB^*)dt$ of a weak solution of stochastic differential equations? [closed]
Set an probability space $(\Omega,{\mathcal F},P)$ with filtration ${\mathcal F}_{t,t \ge 0}$, and
seperatable Hibert space $H$ and $U$
a self adjoint, sectoral，densely defined operator A on H with ...
0
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0
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44
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Commutators of unbounded operators on Hilbert spaces
Commutation seems to be a tricky business when it comes to unbounded operators, because of the domain questions. I have some trouble understanding the usual material about commutators of unbounded ...
- 57
1
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1
answer
60
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Proving an Inequality in Hilbert Space: $\int_0^1 \chi_{[t,\infty)}(T) dt \le T$ for $T\ge 0$
Let $\mathcal H$ be a Hilbert space and $\mathcal B(\mathcal H)$ denotes the set of all bounded operators on $\mathcal H$. An element $T\in \mathcal B(\mathcal H)$ is positive, we write $T\ge 0$ if $\...
- 127
1
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1
answer
62
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Basis of eigenfunction of nonself-adjoint operator
If $H$ is a separable Hilbert space, $A$ is a bounded nonself-adjoint operator, $\{\lambda_n\}_{n\in\mathbb{Z}}$ are the eigenvalues of $A$, and the corresponding eigenfunctions are $\{\psi_n\}_{n\in\...
- 169
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22
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Why is the von Neumann inequality not always fulfilled for n-tuples of commuting contractions? Why cant we just take the single dilations and get it?
Let $(T_1, \ldots, T_n)$ be an $n$-tuple of contractions in a Hilbert space $H$. We know, that for every single $T_i$, there exists a unitary operator $U_i$ in a Hilbert space $K_i$, such that $$T_i = ...
- 41
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1
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45
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Is my formula for this projection correct?
Let $\phi \in L^{2}(\mathbb{R}^{d})$ be fixed. Denote by $P$ the orthogonal projection onto the subspace orthogonal to $\text{span}\{\phi\}$. In other words, for $f \in L^{2}(\mathbb{R}^{d})$ set:
$$(...
- 923
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0
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49
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Lower bound of integral operator in $L^{\infty}$
Let $\mu$ and $\nu$ be two $\sigma$-finite measures, and consider the operator (supposed well-defined) $L^{\infty}(\mu)$ to $L^{\infty}(\nu)$ by $Tg(y) = \int T(x,y)g(x) \mu(dx)$ where the kernel $T(x,...
5
votes
0
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125
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Are creation and annihilation operators special?
In Weinberg's The Quantum Theory of Fields,volume I, the author quotes a theorem that left me a bit mystified. He states
Any operator $O: \mathscr{H} \rightarrow \mathscr{H}$ may be written
$$O=\sum_{...
- 1,724
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2
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45
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Is the resolvent of a local operator local?
Let $A$ denote a bounded linear operator on the Hilbert space $l^2(\mathbb{Z})$. We call $A$ a local operator if and only if there exists a $C \geq 0$ such that $\langle e_x | A | e_y \rangle = 0 $ if ...
- 133
1
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1
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47
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Unique extension of $*$-representation into an abstract multiplier algebra
I'm trying to find a proof of the following fact:
Let $A,B$ be $C^{*}$-algebras and $\pi: A \longrightarrow M(B)$ be a non-degenerate homomorphism in the sense that $\pi(A)B$ densely spans $B$. Then ...
- 1,399
-1
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14
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Bergman projection maps $L^q \left (\mathbb D^2 \right )$ boundedly onto $\mathbb A^q \left (\mathbb D^2 \right )$ for any $q \geq 2.$
Let $\mathbb A^2 \left (\mathbb D^2 \right )$ be the Bergman space consisting of square integrable holomorphic functions on $\mathbb D^2$ and $\mathbb P : L^2 \left (\mathbb D^2 \right ) \...
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