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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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,...
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Index of Callias operator and application in physics
In his article "Axial Anomalies and Index Theorems on Open Spaces" (https://link.springer.com/article/10.1007/BF01202525) C.Callias shows how the index of the Callias-type operator on $R^{n}$...
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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_{...
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Kraus operators
Suppose we have a POVM given by the family of positive, hermitian operators $\{E_i\}_{i\in I} \in \mathcal{H}$.
From the Neimark dilation theorem we know that the given POVM can be obtained from ...
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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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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 ...
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let $\phi : B(l^{p}(X)) \to B(l^{p}(Y)) $ be an isomorphism, does $\phi$ necessarily preserve rank of operators?
let $l^{p}(X)$ and $l^{p}(Y)$ be some $l^p$ function spaces, $B(l^{p}(X))$ and $B(l^{p}(Y)) $ be bounded linear operators on themselves, $\phi : B(l^{p}(X)) \to B(l^{p}(Y)) $ be an isomorphism as ...
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Limiting behavior of integral representation of $(\sqrt{\alpha^2-\partial_x^2}-\alpha)f(x)$
While studying pseudo-differential operators of type $\left(\sqrt{\alpha^{2} - \partial_{x}^{2}}-\alpha\right)\operatorname{f}\left(x\right)$, I came across the following integral representation of ...
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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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Heat semigroup on $C_b(\mathbb R)$
Let $(X,\|\cdot\|)\in \{(L^2(\mathbb R),\|\cdot\|_{L^2}),(L^\infty(\mathbb R),\|\cdot\|_{L^\infty}),(C_b(\mathbb R),\|\cdot\|_{\infty})\}$
I have a question regarding the heat semigroup $$T_tf:=(\...
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About evolution problem with variable coefficients.
I'm studying differential operators. For example, in the evolution equation
\begin{align}
u_t&=(1-\partial_x^2)u\\
u(0)&=u_0
\end{align}
Question 1. Does this problem have any name in the ...
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Mean ergodic Operators [closed]
I am currently listening to some functional analysis lecture and solved the following exercise:
Let E be a Banach space, $T,S:E\rightarrow E$ bounded linear Operators, such that $\exists k \in \Bbb N:...
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Operator exponential equality question: Does $X(\sigma) = Y\sigma$ imply $\exp(X)(\sigma) = \exp(Y)\sigma$?
See this question and other linked questions I am exploring on physics stack exchange. https://physics.stackexchange.com/questions/819663/ad-circ-exp-exp-circ-ad-and-ei-theta-2-hatn-cdot-sigma-sigma-e-...
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Bishop's approximation theorem
I am trying to study the generalization that Axler made to Cuckovic work on the commutants of $T_{z^n}$ (Toeplitz operator with symbol $z^n$) on $L^{2}(\mathbb{D},dA)$, in the resource I am using, the ...
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Injectivity of Kernel Operator in Lp spaces
here is the context: Let $T$ be a kernel operator from $L^1(\mu)$ to $L^1(\nu)$ (probability measures in my problem), defined by
$ (Tf)(x) = \int f(y) \, k(x,y) \, \mu(dy). $
More generally, is there ...