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.
10,436
questions
0
votes
1
answer
30
views
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 ...
- 537
0
votes
0
answers
18
views
$T$ 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^*_{\...
- 607
1
vote
0
answers
38
views
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
votes
0
answers
8
views
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 $c Q_2$, ...
- 1
0
votes
0
answers
31
views
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
vote
0
answers
41
views
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
vote
0
answers
36
views
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
0
votes
0
answers
33
views
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
votes
0
answers
44
views
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
vote
1
answer
60
views
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
vote
1
answer
62
views
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
0
votes
0
answers
22
views
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 $$T_i = ...
- 41
0
votes
1
answer
45
views
Is my formula for this projection correct?
Let $\phi \in L^{2}(\mathbb{R}^{d})$ be fixed. Denote by $P$ the orthogonal projection onto the subspace orthogonal to $\text{span}\{\phi\}$. In other words, for $f \in L^{2}(\mathbb{R}^{d})$ set:
$$(...
- 923
0
votes
1
answer
90
views
Does there exist a widely-used operator $\boxdot$ such that $(\theta \boxdot \phi)(x) := \theta(x) \circ \phi(x)$?
Let $\forall X,Y : L(X,Y)$ symbolize the set of all linear operators from $X \rightarrow Y$.
Let us have operator-valued functions $\theta : I \rightarrow L(Y,Z)$ and $\phi : I \rightarrow L(X,Y)$.
It ...
-1
votes
1
answer
27
views
Stability of Subspaces under a Linear Map in Direct Sum Decomposition
Consider the vector spaces $D_1$, $D_2$, $D$ and $X$ such that $D\subset X$ and $D=D_1\oplus D_2$.
Furthermore, suppose that $L:X\longrightarrow D$ is a linear map such that $D_1$ is stable under $L$...
- 87