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.

learn more… | top users | synonyms

2
votes
1answer
89 views

Is $\text{rk}L=\text{rk}L^*L $ true for finite rank operators?

Let $L$ be a compact linear operator in an infinitedimensional space that has finite rank. Do the equations $$\text{rk}L=\text{rk}L^*L\ \text{and} \ \text{rk}L^*L=\text{rk}R,$$ where $R$ is the ...
5
votes
1answer
428 views

Distinct eigenvalues for integral operator?

Is there some sufficient condition on the kernel $K$ of a (say, finite rank, to simplify) integral operator $$ \mathcal K:f(x)\in L^2(\mathbb R)\mapsto \int K(x,y)f(y)dy $$ so that it has all its ...
22
votes
2answers
505 views

Operators with finite spectrum

Suppose that $T$ is a bounded operator with finite spectrum. What happens with the spectrum of $T+F$, where $F$ has finite rank? Is it possible that $\sigma(T+F)$ has non-empty interior? Is it always ...
0
votes
1answer
168 views

A criterion for convergence in the operator norm

Let $L:H\rightarrow H$ be a continuous linear operator and $R_n:H \rightarrow H$ a sequence of continuous linear operators, where $H$ is a Hilbert space. If the $\sum_{n=1}^{k} R_n$ converge pointwise ...
5
votes
1answer
404 views

What are the Eigenvectors of the curl operator?

The curl operator $\vec\nabla\times\mathbb{1}$ can be written as a skew-symmetric 3x3 matrix $$\mathrm{curl} = \begin{pmatrix}0 & -\partial_z & \partial_y \\ \partial_z & 0 & ...
3
votes
1answer
122 views

Rearranging the spectral theorem

The spectral theorem for selfadjoint compact operators $L$ with infinite range says that $$Lx=\sum_{k=1}^{\infty} \alpha_k \langle x,f_k \rangle f_k, $$ where the $f_k$'s form an orthonormal system ...
3
votes
1answer
467 views

Formula for trace of compact operators on $L^2(\mathbb{R})$ given by integral kernels?

Given an appropriate function $K: \mathbb{R}^2 \to \mathbb{C}$, say continuous of compact support, we obtain a compact operator $T$ on the Hilbert space $L^2(\mathbb{R})$ by the formula $$ (T h)(t) = ...
0
votes
1answer
81 views

Continuity of powers in a Banach algebra.

There are some theorems that say in a unital C* algebra $A$ when one can deduce that the functional calculus of a continuous function f is continuous as map from some subset of $A$ to $A$. In the ...
2
votes
1answer
73 views

Norm of quotients

$\newcommand{\Ker}{\operatorname{Ker}}$ Let $X$ be Hilbert space and let $T:X\to X$ be a bounded operator. Define the operator $S: X/\Ker T \to X/\Ker T$ via $S(x+\Ker T)=Sx+\Ker T$. I can show that ...
0
votes
1answer
95 views

Inversion in a unital C* algebra

Let's say that in a unital C* algebra, we have $b \geq a \geq 0$ and $a$ is invertible. Then $b$ is also invertible. Can we conclude that $a^{-1} \geq b^{-1}$? If so, why? Can any related ...
0
votes
0answers
102 views

Is the closure of a symmetric operator unique?

Let $T:D(T)\rightarrow H$ be a densely defined symmetric operator in a Hilbert space H. The closure $\overline T$ of $T$ is defined as the operator whose graph $G(T)$ is the closure of the graph of ...
4
votes
1answer
252 views

Example of a quasinilpotent operator

Can anybody please give me an example of a quasinilpotent operator $T$, i.e. an operator such that $\sigma(T)=\{0\}$ on $l_2$ such that it has finite dimensional but non-trivial kernel and is not ...
1
vote
1answer
200 views

A specific example of the GNS construction

In an introduction to the GNS construction, I'm told that the GNS construction is a generalization of the way that $L^{\infty} (X, \mu)$ has a representation on $L^2$ where $\mu$ is a measure on $X$. ...
3
votes
1answer
774 views

Compactness and boundedness of integral operator

I got some trouble with my homework question : Let $B$ be the unit ball in $\mathbb{R}^d$, and let $T$ be an integral operatpor on $L^2(B)$ with kernel $K(x,y)$. Suppose that $\sup_x \int_B ...
12
votes
1answer
257 views

Singular-value inequalities

This is my question: Is the following statement true ? Let $H$ be a real or complex Hilbertspace and $R,S:H \to H$ compact operators. For every $n\in\mathbb{N}$ the following inequality holds: ...
3
votes
1answer
146 views

Operator norm of the sum of a finite collection of bounded linear operator

I recently got some difficulty with my homework question. The question is: Let $T_1,\dots,T_N$ be a finite collection of bounded linear operators on a hilbert space $H$, each of operator norm $\le ...
9
votes
2answers
215 views

Are there n-th roots of differential operators?

In analogy to a Dirac operator, it seems to me that formally, the equation $$\frac{\partial^n}{\partial x^n}f(x,y)=D_yf(x,y)$$ is solved by $$f(x,y)=\exp{(x \sqrt[n]{D_y})}\ g(y).$$ Is there a ...
3
votes
1answer
200 views

What are the requirements for a “test” function to show operators commute?

To show that two operators $\hat{A}$ and $\hat{B}$ commute, we can check whether $\hat{A}\hat{B}f(x)$ = $\hat{B}\hat{A}f(x)$. My question is regarding the function $f(x)$. To check that $\hat{A}$ and ...
2
votes
1answer
102 views

If a unital strictly continuous homomorphism $M(A) \to M(B)$ is injective on $A$, is it injective?

Let $A$ and $B$ be C* algebras and let $\varphi : M(A) \to M(B)$ be a unital $*$-homomorphism of their multiplier algebras. Suppose, in addition, that $\varphi$ is strictly continuous (of course, norm ...
3
votes
2answers
415 views

Hellinger-Toeplitz theorem use principle of uniform boundedness

Suppose $T$ is an everywhere defined linear map from a Hilbert space $\mathcal{H}$ to itself. Suppose $T$ is also symmetric so that $\langle Tx,y\rangle=\langle x,Ty\rangle$ for all ...
1
vote
0answers
215 views

Diagonal Dominance and Spectral Radius

For positive semi-definite matrices, $A$ and $B$ with real entries, Let: $X=I-(2Diag(A)-B)^{-1}(A-B)$ The spectral radius $\rho(X) \leq ||X||$. As, $(2 Diag(A)-B)$ becomes a better approximation ...
6
votes
1answer
109 views

A Marcinkiewicz approach

The problem was to prove the following that the operator $$Tf(x)=\int_{\mathbb{R}^N}\frac{f(y)}{|x-y|^\alpha}dy$$ Is continuous from $$L^1 \to \ L_\mathrm{Weak}^{p}$$ where $0<\alpha<N$ and ...
2
votes
0answers
213 views

Eigenfunctions/Invariance of generic convolution operators

Suppose we are given a convolution operator $$ \mathcal{K}[f\,](t):=\int K(t-s)f(s)ds $$ acting on $f\in H_1$ where $H_1$ is a vector space with orthonormal basis $\{\phi_n(t)\}_{n=0}^{N-1}$. If ...
1
vote
1answer
44 views

Vanishing ratio of norms implies vanishing ratio of individual elements?

Consider two vectors $x,y \in \mathbb{R}^n$ be parameterized by a value $t>0$, and suppose that $$\lim_{t \rightarrow 0} \frac{|x(t)|}{|y(t)|}=0,$$ where $|\cdot|$ denotes the standard Euclidean ...
2
votes
1answer
178 views

Is the Neumann series of a real semidefinite matrix asymptotic?

Let $A \in \mathbb{R}^{m \times m}$ be a real symmetric negative-semidefinite matrix and consider the Neumann series $$\sum_{k=0}^\infty t^kA^k = I + tA + t^2 A^2 + \cdots$$ where $t > 0$ is a ...
4
votes
4answers
114 views

Spivak Exercise involving operator norm

The exercise as stated: If $T:\mathbb{R}^{m}\to \mathbb{R}^{n}$ is a linear transformation, show that there is a number $M$ such that $|T(h)|\leq M\cdot |h|$ for all $h\in \mathbb{R}^{n}$. Hint: ...
0
votes
1answer
70 views

Extension of an operator defined on (not necessarily closed) subspaces

Is there always an extension of an operator $T:U\rightarrow W$, defined on (not necessarily closed) subspaces of the infinitedimensional Hilbert spaces $H\supseteq U,L\supseteq W$, to the operator ...
1
vote
1answer
79 views

Can the range of this operator be closed?

Given an operator $T:H\rightarrow L$, where $H$ is a finite-dimensional Hilbert space and $L$ an infinitedimensional one, is the range of $T$, $T(H)$, a closed set in $L$ ? I know that the image ...
6
votes
2answers
179 views

Question about Angle-Preserving Operators

This an exercise out of Spivak's "Calculus on Manifolds". Edit: There was a typo in the exercise as is noted below in the answers. The statement has been edited to reflect this. Given ...
2
votes
2answers
146 views

Why Strongly Continuous Representations?

When working with not-necessarily-finite-dimensional representations, the topology on $GL(V)$ makes a difference. My experience has been that usually people require that the representation $\pi ...
1
vote
2answers
292 views

Norm of the sum of projection operators

Is it true that $$|| a R+b P||\leq\max \{|a|,|b|\},$$where $a$ and $b$ are complex numbers and $P,R$ are (orthogonal) projection operators on finite-dimensional closed subspaces of an ...
1
vote
1answer
45 views

Does this sequence of operators in Hilbert space stop at $\text{rank}T+1$ steps?

Let $\left(T_{n}\right)_{n}$ be a sequence of operator in a infinitdimensional Hilberspace $H$, defined as restrictions of an operator $T:H\rightarrow H$, on smaller and smaller subsets, by the ...
1
vote
2answers
132 views

Cauchy+pointwise convergence $\Rightarrow$ uniform converges (for an operator in a Hilbert space)

Suppose that the sequence of operators in a Hilbert space $H$, $\left(T_{n}\right)_{n}$, is Cauchy (with respect to the operator norm) and that there is an operator $L$, such that ...
0
votes
0answers
146 views

When the linear operator is continuous.

Could I have a hint please on how to prove the following proposition: Let $X$ and $Y$ be two normed space and $T$ be a linear operator from $X$ into $Y$. The operator $T$ is continuous if the ...
2
votes
1answer
115 views

Has this operator $0$ as an eigenvalue / where is my error?

I know of a theorem that tells me, that every compact linear operator on an infinitedimensional Hilbert space has to have the eigenvalue $0$. On the other hand I have the operator \begin{eqnarray*} ...
2
votes
1answer
63 views

How to lift from $B(X)/\mathcal{K}(X)$ to $B(X)$?

I am trying to solve the following problem: $T\in B(X)$. If each $\lambda\in \sigma(T)\backslash\{0\}$ is an isolated point in $\sigma(T)$ and the riesz projection corresponding to $\lambda$ is of ...
6
votes
3answers
444 views

A compact operator in $L^2(\mathbb R)$

Let $g \in L^{\infty}(\mathbb R)$. Consider the operator $$ \begin{split} T_g\colon & L^2(\mathbb R)\to L^2(\mathbb R) \\ & f \mapsto gf \end{split} $$ Prove that $T_g$ is compact ...
8
votes
1answer
267 views

Selfadjoint compact operator with finite trace

I have a compact selfadjoint operator $T$ on a separable Hilbert space. For some fixed orthonormal basis, the operator's diagonal is in $\ell^1(\mathbb{N})$. Can we conclude that $T$ is trace ...
6
votes
2answers
367 views

Why do we distinguish the continuous spectrum and the residual spectrum?

As we know, continuous spectrum and residual spectrum are two cases in the spectrum of an operator, which only appear in infinite dimension. If $T$ is a operator from Banach space $X$ to $X$, $aI-T$ ...
1
vote
0answers
46 views

Similarity orbit of compact operators

I am considering a problem connecting the spectra of compact operators to larger class of operators. Since spectra are invariant under similarity, I wonder whether there is a good reference on ...
4
votes
1answer
186 views

Schwarz inequality for unital completely positive maps

I came across the following form of Schwarz inequality for completely positive maps in Arveson's paper: Let $\delta:\mathcal{A}\to\mathcal{B}$ be a unital completely positive linear map between ...
0
votes
0answers
71 views

Operators with totally disconnected spectra

In their recent paper, Marcoux, Radjavi and Popov show that for $T\in B(\mathcal{H})$, total discontinuity of $\sigma(T)$ implies the existence of almost invariant subspace for $T$. So a natural ...
1
vote
1answer
157 views

Two questions from Dixmier's book on Von Neumann algebras

It seems something is going wrong with the preview I linked in some of my previous questions, so I will just type out the question. I am having trouble with Dixmier's proof of Corollary 5 on p. 46. ...
5
votes
2answers
300 views

Uniform boundedness principle statement

Consider the uniform boundedness principle: UBP. Let $E$ and $F$ be two Banach spaces and let $(T_i)_{i \in I}$ be a family (not necessarily countable) of continuous linear operators from $E$ into ...
2
votes
0answers
298 views

Adjoint of the infinitesimal generator of a stochastic process

I need help seeing that $$ \mathcal{L}^* g = -\frac{\partial (bg)}{\partial x} + \frac{1}{2}\frac{\partial^2(\sigma^2g)}{\partial x^2} $$ is the adjoint operator of $$ \mathcal{L} = b\frac{\partial ...
2
votes
1answer
131 views

A simple question about the open mapping theorem

$X, Y $ : Banach space, $T : X \to Y$ : linear bounded operator, onto. I'm studying open mapping theorem, but how can I prove this? If $B_Y (0, \epsilon_1 ) \subset \overline{T(B_X (0, \epsilon_2 ...
0
votes
1answer
133 views

why isn't “functional operator” a contradiction?

Consider the term "functional operator". My understanding was that: (a) An operator in this context refers to a mapping from one vector space to another vector space. (b) A functional is a mapping ...
5
votes
2answers
185 views

Why are compact operators 'small'?

I have been hearing different people saying this in different contexts for quite some time but I still don't quite get it. I know that compact operators map bounded sets to totally bounded ones, that ...
1
vote
1answer
72 views

What are vector states?

I am reading some papers from the 70s on operator theory. I come across the term 'vector state' of a $C^*$-algebra quite often. It is a little bit confusing. Wikipedia redirects to quantum state ...
0
votes
0answers
88 views

weak closures of ideals [duplicate]

Possible Duplicate: Two questions from Dixmier's book on Von Neumann algebras On p. 46-47 in Dixmier's book on Von Neumann Algebras, which I just realized can be accessed through this ...