Questions tagged [operator-theory]
Operator theory is the branch of functional analysis that focuses on bounded linear operators, but it includes closed operators and nonlinear operators. Operator theory is also concerned with the study of algebras of operators.
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Boundedness of a matrix operator in a norm
I would like to ask a simple question.
How do I show that a matrix is a bounded linear operator, for example this matrix
$$A=\begin{bmatrix}
t~~0\\
0~~\frac{1}{\sqrt{t}}
\end{bmatrix}$$
I know that ...
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$T$ compact operator, Let $\Delta^*_{\bar{\lambda}}$ subspace of $X^*$. Prove $\Delta^*_{\bar{\lambda}} = \bar{\Delta^*_{\bar{\lambda}}}$
We have proved the following claim:
Let T be compact operator and $\lambda \neq 0$. Then $\Delta_\lambda = \bar{\Delta _\lambda}$.
Now there is the corollary:
$T$ compact operator, Let $\Delta^*_{\...
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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 $x \in$ ...
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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{...
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Almost orthogonal operators after a relative scaling
If two positive operators $Q_1$ and $Q_2$ with unit $\ell_1$ norm are almost orthogonal: $\parallel Q_1 - Q_2 \parallel_1 \geq 2 -\epsilon$, then what can we say about the operators $Q_1$ and $c Q_2$, ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $\...
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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\...
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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 $$T_i = ...
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Schwarz inequality for unital completely positive maps
I came across the following form of Schwarz inequality for completely positive maps in Arveson's paper:
Let $\delta:\mathcal{A}\to\mathcal{B}$ be a unital completely positive linear map between two $...
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Is my formula for this projection correct?
Let $\phi \in L^{2}(\mathbb{R}^{d})$ be fixed. Denote by $P$ the orthogonal projection onto the subspace orthogonal to $\text{span}\{\phi\}$. In other words, for $f \in L^{2}(\mathbb{R}^{d})$ set:
$$(...
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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 ...
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