# Tagged Questions

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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### Problem involving the Spectral Mapping theorem.

Consider the following problem: Let $T$ be a bounded operator in a Banach space $X$. Use the Spectral Mapping theorem to show that $|\lambda^n|\le\|T^n\|$ for all $\lambda\in\sigma(T).$ Here's ...
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### Consider the Banach Space $C[0,1]$. Find decomposition of spectrum of the indefinite integral operator.

Cosider the Banach Space $C[0,1]$ of real-valued continuous function on $[0,1]$ with the supremum norm. and the linear operator $$A: x(t)\mapsto\int\limits_0^tx(s)ds$$ Find its eigenvalues, ...
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### Existence of Unitary Map

I've been recently introduced to Unitary operators of a Hilbert space and I've been wondering the following. Existence of a unitary operator $T$ on a (possibly infinite) Hilbert space $H$ is simple ...
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### Positive bounded operators

Let $A,B$ be positive self-adjoint bounded operators and $\lambda >0$ then I want to show that if $$A-B \ge 0$$ that is $\langle x,(A-B)x \rangle \ge 0$ we have that the resolvents (whose ...
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### prove that all pure states in a commutative C* algebra are multiplicative linear functionals

I am trying to prove this , but can not see it clearly. it was given as some sort of converse of the fact that all multiplicative linear functionals are pure states
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$A:\cal{P}_1 \to \cal{P}_1$ is a linear operator defined with $$A(p)(t):=(3t+1)p'(t)+2p(t).$$ I'm trying to find a basis $e$ of $\cal{P}_1$ in which the operator $A$ has the matrix form $$A= ... 1answer 60 views ### Find the eigenvalues of the operator T. I have the following problem, "Suppose that X=\ell^1 and define the operator T\in B(X) as follows:$$Tx=\left(\frac12x_2,\frac13x_3,\frac14x_4,...\right)\,,\textit{where,}\,\,\, ...
Let $X$ be a Banach space (it is in fact an $L^p$ space) and let $T:X \to X$ be a linear continuous operator (which is not injective and not surjective). I am trying to figure out if the following is ...