# 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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### Representing Vector Norm Through Operator Norms

Let $V$ be a vector space over $\mathbb R$ and $v\in V$. We endow $V$ with a norm $\|\cdot\|_V$ and $\mathbb R$ with the standard norm. I am trying to prove \sup \{\lvert Lv\rvert \...
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### Non-injective closure of injective unbounded operator [duplicate]

Let $A\colon H\to H$ be an unbounded operator on a Hilbert space $H$ with domain dom$(A)$. Suppose that $A$ is injective. Suppose that $A$ is closable (or symmetric, even). Question: Does this imply ...
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### Nonunital weak operator closed self adjoit subalgebra has a projection that acts like an identity operator?

Let $A$ be weak operator closed self adjoint subalgebra of the bounded operators for a Hilbert space $H$(may not have identity so cannot use usual von neumann alegbra properties). I have shown that if ...
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### Does this integral operator between Banach spaces have a non-trivial kernel?

In a question which was recently asked, the goal was to (dis)prove that the set of functions which satisfy a certain equation is trivial. There is already a very neat solution there, but I came up ...
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### Applying Spectral Mapping Theorem to determine if $f(T)$ is compact

Given a self-adjoint, compact operator $T$ and a continuous $f:\sigma(T)\to\mathbb{R}$, I'm trying to determine the conditions under which $f(T)$ is also compact. I know of the Spectral Mapping ...
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### Some questions about integration and operator theory.

As we all know there are multiple integral operators which all basically do the same thing in various contexts. I am talking about operators like the Lebegues integral, Riemann integral and more ...
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### Finding the conjugate operator of the following operator

Let $A$ be linear operator from $l^2$ to $l^2$ such that $Ax =y^0 \cdot \sum_{1}^{\infty}{x_k}$ , where $y^0 \in l^2$ — fixed element. Show, that conjugate operator $A^*$ exists and find it. Show, ...