# 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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### If $T$ has finite rank, then: $I-T$ is injective if and only if $I-T$ is surjective?

I have a Banach Space $X$ and an linear continuous operator $T\colon X\to X$ that has finite rank (i.e. $\dim {T(X)}<\infty$). Then, $I-T$ is injective if and only if $I-T$ is surjective?
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### An hermitian operator problem

It is possible to have two hermitian operators $A$ et $B$, with : $B^2 = \mathbb{I}d$ $[A,B] = i * \mathbb{I}d$ where $i$ is the usual (complex) square root of $(-1)$, and $\mathbb{I}d$ is the ...
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### For positive invertible operators $C\leq T$ on a Hilbert space, does it follow that $T^{-1}\leq C^{-1}$?

I need the following result. I think it's quite obvious but I don't know how to prove that: Let $C, T : \mathcal{H} \rightarrow \mathcal{H}$ be two positive, bounded, self-adjoint, invertible ...
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### Reflexive Banach algebras?

I have been reading Gelfand theory for a while and it just occurs to me that the whole theory is an analogy to what we did for Banach spaces. For a Banach space $X$, we investigate its dual $X'$ ...
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### The union of cyclic subspace is also a cyclic space

Given a separable Hilbert space $H$, $U$ is a unitary operator. A cyclic subspace, denoted as $Z(x)$ for some $x\in H$, is defined as the closure of linear span of $U^nx$, where $n\in \Bbb Z$ is any ...
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### Compute an operator norm

Consider the operator $M$ acting on the space $\mathbb{R}[X]$ of real polynomials by $Mp(x)=xp(x)$. We equip $\mathbb R[X]$ with the $L^2$ norm $$\|p\|^2=\int p(x)^2d\mu(x),$$ where $\mu$ is a ...
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### Biduals of operators

Let $T\colon X\to Y$ be a bounded linear operator acting between Banach spaces. Suppose $T$ is an isomorphism onto its range. Must $T^{**}\colon X^{**}\to Y^{**}$ be an isomorphism onto its range also?...
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### Closure of the invertible operators on a Banach space

Let $E$ be a Banach space, $\mathcal B(E)$ the Banach space of linear bounded operators and $\mathcal I$ the set of all invertible linear bounded operators from $E$ to $E$. We know that $\mathcal I$ ...
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### A complete eigenvector basis for the restricted operator

Let $X$ be a (not necessarily bounded) selfadjoint linear operator on a Hilbert space $H$ and let $M$ be a closed subspace such that $X(M) \subset M$. Suppose that $X$ admits an orthonormal basis ...
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### Image of a Markushevic basis

Let $X, Y$ be Banach spaces and let $T\colon X\to Y$ be a bounded linear operator. Assume that $X$ admits a Markushevic basis. Does $\overline{T(X)}\subseteq Y$ admit a Markushevic basis as well? ...
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### Bounded operator from a Hilbert space to $\ell^1$ is compact

Let $H$ be any Hilbert space. How can we prove that any bounded linear operator $T\colon H \to \ell^1$ is compact? If we use the fact that the space $\ell^1$ has Schur property (norm and weak ...
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### On Pitt's theorem

The famous Pitt's theorem asserts that if $p>q$ then each bounded operator $T\colon \ell^p\to\ell^q$ is compact. Since $\ell^p$ and $\ell^q$ are incomparable ($p\neq q$, $p,q\geq 1$), each operator ...
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### Existence of faithful state in $C^\ast$-algebras

Why does there always exist a faithful state in a separable $C^\ast$-algebra?
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### Compact operators on an infinite dimensional Banach space cannot be surjective

I am reading a book about functional analysis and have a question: Let $X$ be a infinite-dimensional Banach-space and $A:X \rightarrow X$ a compact operator. How can one show that $A$ can not be ...
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### Unitary Operator as a complex valued function

A book on Quantum Mechanics by Schwinger states, "A unitary operator can be considered to be a complex valued function of a Hermitian operator." Please give a hint on how to prove this assertion.
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### Left regular representation of $L^1(G)$ for a locally compact group $G$

Let $G$ be a locally compact group (not discrete) and let $L$ be the left regular representation of $A = L^1(G)$ on itself i.e. $L: A \to \mathcal{B}(A)$ where $L(f): A \to A$, $L(f)(g) = f*g$. I want ...
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### For a multiplication operator $M_f$ on $L^2$ with $f\geq 0$, is $SM_fS^{*}$ positive?

I have the following problem. Let $\Omega \subset R^n$ have finite measure, let $H = L^2(\Omega)$ and let $S: H \to H$ be a bounded linear operator. Then it is well known that $P = SS^*$ is a positive ...
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### The inequality with a differential operator

Let $A f = a f'''$ for $f\in C_0^3(\mathbb R)$ where $a$ - some constant. Is it possible to find $a$ such that $$\|\lambda f-A f\|\geq \|\lambda f\|$$ for all $f\in C_0^3(\mathbb R)$ ...
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When is the following operator self-adjoint?(Is there a difference between self-adjoint and Hermitian?) $O:= \sum_{n=0}^4 f_n(x){d^n\over dx^n}$ subjected to boundary conditions $y(0)=y'(0)=y(1)=... 3answers 2k views ### Square root of differential operator If$D_x$is the differential operator. eg.$D_x x^3=3 x^2$. How can I find out what the operator$Q_x=(1+(k D_x)^2)^{(-1/2)}$does to a (differentiable) function$f(x)$? ($k$is a real number) For ... 1answer 207 views ### Is a projection operator diagonal-decreasing on a positive operator? I just faced a obvious-looking inequality, but I didn't manage to prove it. Let$H$be a finite-dimensional Hilbert space,$M, \rho$positive operators on$H$,$P$an orthogonal projector on$H$. Is ... 0answers 454 views ### Sturm-Liouville Theorem I was reading the Wiki page on the Sturm-Liouville theory. Why are those tenets true? Are there any (not too advanced) reference material? I have also read that "There are countably infinite ... 1answer 698 views ### Eigenvalues, kernel and rank of a compact operator: how to start? I'm trying to solve the following exercise: Let$f\in\mathcal{C}([0,1])$and let$T$an operator such that$Tf(x)=\int_0^1(x-t)f(t)dt$. I have proved that$T$is a bounded linear operator and, by ... 0answers 306 views ### Construct a multiplication operator which has dense point spectrum By a multiplication operator here we mean an operator $$Af(t)=m(t)f(t), \qquad f \in D(A)=\{x \in L^2(\mathbb{R} \mid m(t)f(t) \in L^2(\mathbb{R})\}$$ where$m$is a Borel measurable function on ... 0answers 855 views ### Continuous spectrum can shrink to an isolated point Let$A$be a bounded linear operator in a Hilbert space$H$. I had the misconception that the continuous spectrum of$A$would necessarily have some "continuous" appearance: an interval, a union of ... 1answer 284 views ### The commutator subgroup of the group of bounded invertible linear operators I am curious to know what the commutator subgroup of the group of (bounded) invertible linear operators on a complex Hilbert space is? Note that by "commutator subgroup" I mean the subgroup ... 2answers 390 views ### Injection from non-separable to separable subspaces Let$\Gamma$be an uncountable set (possibly of cardinality$\aleph_1$). Is there an injective bounded linear operator$T\colon c_0(\Gamma)\to X$, where a)$X$is some separable Banach space b)$X=...
Consider a square matrix $P$. We call it stochastic if it holds that $$p_{ij}\geq0\text{ and } \sum\limits_{j=1}^m\,\,\,\,p_{ij} = 1$$ for all $1\leq i,j\leq m$. I wonder when the following limit ...
How to prove Halmos’s Inequality? If $A$ and $B$ are bounded linear operators on a Hilbert space such that $A$, or $B$, commutes with $AB-BA$ then $$\|I-(AB- BA)\|\ge 1.$$ I found it from http://...