# 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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### Show that operator norm of linear map is equal to $a^2$

Let $a>0$. Show that the linear map $A$ that assigns every $x \in C([0,a])$ the function $s \mapsto s \int_0^a x(t) dt$ from $C[0,a]$ has operator norm $a^2$. I'm not really sure where to start ...
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### Given $[A,B]=0$ then $[A,f(B)]=0$

Given $A$ and $B$ two operators that commute ($[A,B]=0$) then $A$ commutes with an arbitrary function of $B$ I recently saw this property of the commutators on a quantum mechanics course. We didn't ...
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• 943
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### Showing 1 cannot be an eigenvalue after rotating an operator

I have a compact self-adjoint positive integral operator $Q:L^2(0, \infty) \to L^2(0, \infty)$ with operator norm $\| Q\| =1$. By the assumptions, we know $1$ is an eigenvalue of $Q$. Let $y\ne 0$ and ...
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### How much can the (reduced) group $C^*$ algebra maintain the subgroup structure?

I have read a theorem which says that if $H$ and $G$ are both discrete groups, $G\leq H$, then $C^*(G)$ is a subalgebra of $C^*(H)$ and $C^*_r(G)$ is a subalgebra of $C^*_r(H)$. For now, what I can ...
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For every $n \in \mathbb{N}$ let $T_n:c_0 \rightarrow \mathbb{R}$ be defined through $T_n(x_k)_k=x_n$. First, I need to show that $T_n$ is bounded, which is the case if and only if $T_n$ is continuous....
### Strong limit from the unitary orbit of $A$
Let $A,X\in L(H)$ with $\Bbb D\subset W_e(A)$ ,where $W_e(A)$ is the essential numerical range of $A$,and $\|X\|\leq 1$. Then there exists a sequence of unitaries $(U_n)_n$ in $L(H)$ such that wot \$...