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.

Filter by
Sorted by
Tagged with
5
votes
2answers
359 views

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?
4
votes
2answers
553 views

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 ...
6
votes
1answer
1k views

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 ...
3
votes
1answer
423 views

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 $X'$ ...
4
votes
1answer
186 views

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 ...
2
votes
1answer
356 views

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 ...
4
votes
1answer
99 views

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?...
15
votes
2answers
2k views

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 $\mathcal I$ ...
6
votes
1answer
172 views

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 ...
1
vote
1answer
80 views

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? ...
4
votes
2answers
144 views

Weak convergence as convergence of matrix elements

Let $H$ be a Hilbert space with orthonormal basis $(e_h)_{h \in \mathbb{N}}$ and let $(A_n)_{n \in \mathbb{N}}$ and $A$ be bounded linear operators. We say that $A_n$ converges weakly to $A$ if $$\...
4
votes
0answers
292 views

Neumann series in an incomplete normed algebra

Let $\mathcal{A} \equiv (A, \|\cdot\|_A)$ be a unital (associative) normed algebra over the real or complex field, and assume that $\mathcal{A}$ is not complete. Provided $\mathcal{B}_\mathcal{A}$ is ...
4
votes
1answer
223 views

Powers of a densely-defined bounded linear operator

This is a question I was thinking of some time ago. Suppose $\mathbf{X} \equiv (X, \|\cdot\|_X)$ is a (real or complex) Banach space, $U$ is a dense subspace of $\mathbf{X}$, and $\phi$ is a bounded ...
1
vote
1answer
221 views

When is a compact operator differentiable?

When is it possible to prove that a compact operator $T: V \to V$ where $V$ is a Banach space is also differentiable? Fréchet differentiable? PS: There is a further information which might help. My ...
2
votes
0answers
100 views

Extension of Uncertainty Relations to a specific potential in Schrödinger Equation

Given some $\|\psi \|$ $\in$ $L^2 (\mathbb R^n) $ such that $\| \psi \|_2 =1$ and a function (potential) $V: \mathbb R^n \rightarrow \mathbb R$. The Schorödinger equation tells us that $-\triangle \...
1
vote
0answers
129 views

How can I compare unbounded linear operators?

Let $X$, $Y$ be Hilbert spaces. Let $S, T : X \rightarrow Y$ be unbounded operator. Suppose $S$ and $T$ be bounded operators. Then we can compare by their maximum distance on the unit ball of $X$. ...
3
votes
1answer
93 views

Algebraic description of coisometries

Let $H$ be a Hilbert space. An operator $T\in\mathcal{B}(H)$ is called coisometric if it maps open unit ball of $H$ onto open unit ball of $H$. Please tell me how to prove that condition $T$ is ...
3
votes
0answers
196 views

Find spectrum of $Af (x) = \int\limits_0^\pi \sum\limits_{n=1}^\infty 5^{-n} \cos(nx)\cos(nt) f(t)\;dt$

I need to find spectrum of $Af (x) = \int\limits_0^\pi \sum\limits_{n=1}^\infty 5^{-n} \cos(nx)\cos(nt) f(t)\;dt$ in $L_2[0,\pi]$. I know that this operator is self-adjoint, so its residual spectrum ...
3
votes
0answers
775 views

Simple isolated eigenvalue and pole of the resolvent

Let $T$ be bounded linear operator on some complex Banach space, and $\lambda$ an eigenvalue of $T$ which is isolated in its spectrum, and such that $\bigcup_{n\ge 1} N((T- \lambda I)^n)$ is one-...
1
vote
1answer
179 views

Necessary and sufficient condition of being dissipative

I want to know a necessary and sufficient condition on $m:\Omega \mapsto\mathbb{C}$ such that the multiplication operator $M_{m}$ is dissipative in $L^{p}(\Omega)$, where $\Omega$ is a Banach space. ...
1
vote
1answer
93 views

Maximizing $\|APBPA\|_2$ subject to $0 \leq P \leq I$

Given positive semidefinite matrices $A,B$, how to compute $$\max_{0 \leq P \leq I}\|APBPA\|$$ where the norm is the spectral norm, i.e., the largest singular value?
7
votes
3answers
505 views

Nonexistence of a cyclic vector for a representation on $\ell^2(\mathbb{Z})$

Let $S$ be the bilateral shift on $\ell^2(\mathbb{Z})$ and let $T = S + S^*$. I want to show that there is no cyclic vector for the representation of $T$ on $\ell^2(\mathbb{Z})$ i.e. $\forall x\in \...
9
votes
1answer
1k views

Prove that $\sigma(AB) \backslash \{0\} = \sigma(BA)\backslash \{0\} $

Prove that $\sigma(AB) \backslash \{0\} = \sigma(BA)\backslash \{0\} $. Where $A,\ B$ are bounded operators in Banach space and $\sigma$ denotes spectrum.
2
votes
1answer
753 views

A question on linear transformation

Let $T$ be a linear transformation on $\mathbb{R}^{4}$ whose standard matrix is $$\left(\begin{array}{rrrr} 1 & -1 & -1 & -1\\ 1 & 1 & 1 & -1\\ 1 & 1 & -1 & 1\\ ...
6
votes
1answer
600 views

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 ...
11
votes
1answer
669 views

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 ...
4
votes
1answer
848 views

Existence of faithful state in $C^\ast$-algebras

Why does there always exist a faithful state in a separable $C^\ast$-algebra?
1
vote
1answer
904 views

Definition of (formally) self-adjoint

Does formally self-adjoint imply self-adjoint and vice versa? Thanks.
6
votes
3answers
3k views

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 ...
2
votes
2answers
340 views

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.
4
votes
1answer
459 views

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 ...
1
vote
1answer
468 views

Ascent and descent for a bounded linear operator

Let $T$ be a bounded linear operator on some complex Banach space. We define its ascent by $\alpha(T) = \min \{ n \ge 0 \, / \, N(T^n) = N(T^{n+1}) \}$ and its descent by $\delta(T) = \min \{ n \ge 0 \...
2
votes
0answers
166 views

Embedding $\ell^\infty(\Gamma)$ into $\mathcal{B}(E)$

Is there any criterion answering the question: Let $E$ be a Banach space. When does the Banach space $\mathcal{B}(E)$ of all bounded operators on $E$ contain a copy of $\ell^\infty(\Gamma)$? Here $\...
7
votes
2answers
389 views

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 ...
8
votes
1answer
2k views

Hilbert Schmidt integral operator

Hilbert-Schmidt Integral operators are usually defined from $H=L_2[a,b]$ into $H=L_2[a,b]$ as $$(Tf)(x) = \int_a^b K(x,y)f(y) dy,$$ provided that $K(x,y)$ is a Hilbert Schmidt kernel, namely $$\...
3
votes
2answers
479 views

Spectrum of a “quasi” right shift operator

Let $\mathcal{H}$ be a Hilbert space and let {$e_j$}$_{j\in \mathbb{Z}}$ be an orthonormal basis for $\mathcal{H}$. Define a linear operator $T$ on $\mathcal{H}$ by $T(e_0) = 0$ and $T(e_j) = e_{j+1}$ ...
11
votes
1answer
555 views

Renorming $\mathcal{B}(\mathcal{H})$?

Consider the Banach space of all bounded operators $\mathcal{B}(\mathcal{H})$ on a (separable if you wish) Hilbert space $\mathcal{H}$ with the operator norm. Can we renorm this space to a strictly ...
6
votes
1answer
313 views

Invertibility of Toeplitz operator in $\ell_1$

Suppose we have a Toeplitz operator $$ T(a) = \begin{bmatrix} a_{0} & a_{-1} & a_{-2} & \ldots & \ldots &a_{-n+1} &\dots \\\\ a_{1} & a_0 & a_{-1} & \ddots & ...
1
vote
2answers
78 views

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 $f\in C_0^3(\mathbb R)$ ...
0
votes
1answer
247 views

When is this Self-adjoint?

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)=...
8
votes
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 ...
4
votes
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 ...
1
vote
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 ...
6
votes
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 ...
3
votes
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 ...
3
votes
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 ...
2
votes
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 ...
2
votes
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=...
2
votes
3answers
424 views

The limit of matrices

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 ...
10
votes
1answer
449 views

How to prove Halmos’s Inequality

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://...