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

1
vote
0answers
44 views

Approximating an element in the domain of an unbounded operator by a sequence in a dense subset of the domain.

Let $T$ be a closed unbounded (in my case also symmetric) operator on a Hilbert space $\mathcal{H}$ with dense domain $\mathcal{D}(T)$, and let $f\in \mathcal{D}(T)$. Suppose there is a dense ...
1
vote
0answers
26 views

Operator system of minimal dimension with one dimensional projections

Consider the matrix algebra $\mathbb{M}_n(\mathbb{C})$ with H-S inner producr ($\langle a, b\rangle =tr (a^*b)$). What is the minimal dimension of any operator system $\mathcal{A}$ in ...
1
vote
0answers
12 views

Matrix monotone operators Intuition

can anyone explain by intuition that a matrix(operator) $A$ is monotone? I know for normal functions if a matrix is monotone this means intuitively i can think of it as increasing, but hard to ...
1
vote
0answers
39 views

Resolvent set/operator

Just a question here. Why do we study Resolvent operators and resolvent sets? Will there be any motivation or intuition behind this?
1
vote
0answers
35 views

Is the following differential operator closed (closabe)?

Let $L$ be the following differential operator. $L: C^2(\Bbb{R}^2_+)\to C^0(\Bbb{R}^2_+) $ $$Lf = \partial_x f(x,y) (y-x) + \partial_y f (x,y)(x-y) + \frac{1}{2} \bigg( \partial_{xx} f(x,y)x + ...
1
vote
0answers
52 views

Integration of $A$-valued functions (Functional Analysis)

Premise 1: my source is the Rudin - Functional Analysis. Premise 2: i'm not a mathmo so forgive for the mistakes A couple of question on the subject... An example of Banach Algebra is the set of ...
1
vote
0answers
65 views

Eigenvalues of a certain product of matrices with special structure

Sorry for cross-posting from MO. Let $d$ and $c$ be positive integers and $q = dc$. Let $G$ be a $q$-by-$q$ positive semi-definite real matrix with eigenvalues all $\le 1$, and define the ...
1
vote
0answers
39 views

Square root of Self-Adjoint Operator

I have $H=L^2(0,2)$ and $Aw=w^{(4)}, D(A)=H^4(0,2)\cap H^2_0(0,2)$, this operator is non-negative and self-adjoint because $A$ is monotone maximal, $R(I+A)=H$. A form $t(u,v)=(Au,v)_{H}$ is closable, ...
1
vote
0answers
37 views

Root Square Convergence

I'm trying to solve this problem: Prove that if $A_n\geq 0$, $A_n\rightarrow A$ in norm, then $\sqrt{A_n}\rightarrow\sqrt{A}$ in norm. Where everything are bounded and linear operators in a Hilbert ...
1
vote
0answers
25 views

Matrix whose eigenvectors are Hermite polynomials

I first constructed a symmetric matrix as the Laplacian operator, and its eigenvectors are a series of harmonics functions as expected. I programmed it and convinced myself. The matrix looks like: $$ ...
1
vote
0answers
32 views

Finding Linear Operator for a given Basis

Consider a linear operator $$L: \lbrace f: \Bbb{R}\rightarrow \Bbb{R} \rbrace \rightarrow \lbrace f: \Bbb{R}\rightarrow \Bbb{R} \rbrace $$ For example $$ L(f) = f(x+1) - f(x)$$ Define the ...
1
vote
0answers
28 views

Polar decomposition theorem for symplectic and orthogonal Banach Lie groups in infinite dimensional settings

Could you please help me to understand the polar decomposition theorem for $Sp(H, J_Q)$ and $O(H,J_R)$ where $H$ is infinite dimensional separable Hilbert space and $J_R$ and $J_Q$ stands for ...
1
vote
0answers
17 views

Weakly convergent subsequence under continuous operator

Suppose we have two Hilbert spaces $H_1,~ H_2$, a linear continuous operator $T:H_1 \to H_2$ and a weakly convergent sequence $u_k\rightharpoonup u$ in $H_1$. Is $Tu_k \rightharpoonup Tu$ in $H_2$ ...
1
vote
0answers
27 views

(bounded linear) orthogonal projections on Hilbert spaces

If $H$ is a Hilbert space and $T:H\to H$ bounded and linear, such that $T$ is an othogonal projection (i.e. $T^*=T^2=T$), is then T always zero on $im(T)^\perp$ and the identity on it's image, ...
1
vote
0answers
24 views

Substitution from the left and right?

I am a bit confused about this substitution from the left or from the right thing. How does one determine whether to substitute from the left or from the right? And does it even matter as long as i ...
1
vote
0answers
66 views

Arzelà–Ascoli theorem for operators

If you have a net of continuous linear operators between reasonable spaces (complete, at least), does there exist Arzelà–Ascoli-like theorems giving convergent subnets? I believe that it should be ...
1
vote
0answers
32 views

Proposed proof of operator theory result

Hi I am interested in checking my proposed solution to the following problem in Operator Theory: Please give me hints as to how to improve the proposed proof rather than the full correct solution. ...
1
vote
0answers
38 views

Sobolev “unit ball” compact in $L^p$

We consider $I=(0,1)$, and $1< p \leq \infty$. If $B = \{ f \in W^{1,p}(0,1); ||f||_p + ||f'||_p \leq 1 \}$, how to show that $B$ is compact in $L^p(0,1)$?
1
vote
0answers
25 views

Daletskii-S.Krein formula proof

I've came across to the following equation, known as Daletskii-S.Krein formula. Consider a sufficiently smooth function $h : \mathbb{R} \rightarrow \mathbb{R}$, and let $\mathbf{A}_t = \mathbf{A} + ...
1
vote
0answers
78 views

If a linear operator between two normed linear spaces is continuous at one point, then it is continuous at all points.

Let $f : \langle V_1, \|\cdot\|_1\rangle \to \langle V_2, \|\cdot\|_2\rangle$ be linear. Then if $f$ is continuous at some $v \in V_1$, then it is continuous on all of $V_1$. Without appealing to ...
1
vote
0answers
37 views

Jordan normal form

Let $H$ a Hilbert space and let $T\in B(H)$ a bounded operator on H, my question is if it exist a theorem about some "decomposition" of type Jordan canonical form in a general Hilbert space, and how ...
1
vote
0answers
58 views

Find the iverse of the followning bounded operator?

The following definition and Theorem are given in the book "A short course on operator semigroup" by the author "K-J Engel and R Nagel". Sectoral operator: A closed linear operator $(A,D(A))$ in ...
1
vote
0answers
27 views

How to solve differential equations for linear operators?

I want to solve the differential equation $$ BA = \frac{\partial}{\partial t} A $$ for $A$. Here $A : H_1 \mapsto H_2$ and $B : H_2 \mapsto H_2$ are operators and $H_1, H_2$ are some Hilbert spaces. ...
1
vote
0answers
26 views

Abelian Algebras: Generator

Given a Hilbert space $\mathcal{H}$. Consider normals: $$N:\mathcal{D}(N)\subseteq\mathcal{H}\to\mathcal{H}:\quad N^*N=NN^*$$ Denote their algebra: ...
1
vote
0answers
27 views

Conditions for an operator on a Hilbert space to have an orthonormal set of eigenfunctions

I'm working on a problem that requires the following operator, $A^TA$, to have an orthonormal set of eigenfunctions. Note $A:H_1 \mapsto H_2$, where $H_1$ and $H_2$ are separable Hilbert spaces. ...
1
vote
0answers
28 views

Norm of a product of projections in a Banach space

Let $X$ be a Banach space and let $P_1,P_2$ be two projections in $B(X)$, i.e., $P_1^2 = P_1, P_2^2=P_2$. My question: under what conditions do we have that $\Vert P_1 P_2 \Vert = \sqrt{\Vert P_2 ...
1
vote
0answers
54 views

Prob. 6, Sec. 4.3 in Kreyszig's Functional Analysis Book: What are all possible extensions?

Here's Theorem 4.3-2 in Introductory Functional Analysis With Applications by Erwine Kreyszig: Let $X$ be a normed space, let $Z$ be a subspace of $X$, and let $f$ be a bounded linear functional ...
1
vote
0answers
70 views

On the square root function of matrices

Let $A, B$ be positive definite matrices and let $P$ be an orthogonal projection. If $A \leq PBP,$ does it follow that $$ A^{1/2} \leq PB^{1/2}P?$$
1
vote
0answers
26 views

How to extend formula for residue to functional calculus of operators

Suppose $\{X_t\}$ is a stochastic process with the covariance operator $\Gamma$ and the first $d$ eigen values are $\lambda_1\geq\lambda_2\geq \ldots \geq\lambda_d$ with eigen vectors ...
1
vote
0answers
55 views

Frechet derivative of an operator

Let an operator $T:C[a,b]\to C[a,b]$ be defined as: \begin{equation} (Tu)(x)=\int_{a}^{b}K(x,t)f(t,u(t))dt \end{equation} where $K:[a,b]\times[a,b]\to \mathbb{R}$, $f:[a,b]\times \mathbb{R}\to ...
1
vote
0answers
51 views

Generating fractional taylor series

I was considering the notion of taylor series which posit that the sum $$ \sum_{i=0}^{\infty} \frac{1}{i!} a_ix^i $$ Where: $$ a_i = \frac{d^if}{dx^i}_{x= a} $$ Converge to the function f in a ...
1
vote
0answers
35 views

Riesz representation theorem for $\langle\mathcal A u,v\rangle$.

Let $V$ be a Hilbert space, and let $V^*$ denote its dual space, consisting of all continuous linear functionals from $V$ into the field $\mathbb R$ or $\mathbb C$. If $x$ is an element of $V$, then ...
1
vote
0answers
38 views

Eigenvalue-eigenvector equation for an operator

Proof: Given an eigenvalue-eigenvector equation, suppose that the state vector depends on an external parameter, e.g. time, and that over it acts an operator that is the fourth derivative w.r.t. ...
1
vote
0answers
34 views

Purely nondeterministic weakly stationary processes

I found a necessary and sufficient condition for a stochastic process being purely nondeterministic in Ihara (1993). As follows: A weakly stationary process $X$ is purely non-deterministic if and ...
1
vote
0answers
39 views

Spectra of operator matrices

Suppose we are given a bounded linear operator $A\colon X\to X$ on a Banach space which is injective and has closed range. Can we find two other operators $T$ and $S$ say such that ...
1
vote
0answers
58 views

Prob. 9, Sec. 3.10 in Kreyszig's functional analysis book: The image of ann isometric non-unitary operator on a Hilbert space

Let $H$ be a Hilbert space, let $T \colon H \to H$ be a linear operator such that $T$ is isometric but not unitary. Then how to show that the image $T[H]$ is a proper closed subspace of $H$? My ...
1
vote
0answers
28 views

What are these algebraic properties called?

Suppose $O$ is some operator, suppose $f$ and $g$ are both functions, then linearity implies that: $O(\alpha f + g) = \alpha O(f) + O(g)$ What about the following property: $(O_1+O_2)(f) = O_1(f) ...
1
vote
0answers
28 views

The spectral projections of convolution operator

Given a self-adjoint operator $A$ in a Hilbert space $H$. How can one find its spectral projections $\{E_{\lambda}\}_{\lambda\in\sigma(A)}$? In particular, given a convolution operator on $L^2(G)$, ...
1
vote
0answers
64 views

Eigenvalues and eigenvectors of certain diagonal constant matrices

Suppose I have an infinite complex diagonal constant (Toeplitz) matrix, which is also Hermitian. This is given by finite number of complex parameters $z_1, z_2, \cdots, z_k$. If, $z_1$ is the ...
1
vote
0answers
25 views

A question on bounded operator $\|T_A x\|\leq K\|x\|$

We consider an operator $T_A:H\rightarrow H$, where $H$ is an Hilbert Space and $A$ is its associated $N\times N$ matrix. $T_A$ is said "bounded" if there exists a constant $K>0$ such that $$\|T_A ...
1
vote
0answers
25 views

Prove that matrix $[S]$ associated to operator is such that $A |\zeta|^2\leq s_{ij}(x) \zeta_i \zeta_j\leq B |\zeta|^2$.

Let us consider $N\times N$ matrix $[S]$ associated to operator $S:V\rightarrow V$ where $V$ is a Hilbert space; $S$ is linear, bounded, invertible, positive and self-adjoint. Prove that $[S]$ is ...
1
vote
0answers
25 views

Showing that an operator is bijective

Assume that $ A $ generates a contraction semigroup on a Hilbert space $ X $, and B is a bounded linear operator on $ X $. I want to show that $ A + B - 2|| B ||I $ with the domain equal to the domain ...
1
vote
0answers
46 views

Prove that norm of Yosida approximations blows up

I need an help with the following exercise about Yosida approximations. Let's fix the notation: let $A$ be an unbounded linear operator define on a dense domain $D(A)\subset X$ of a Banach space $X$. ...
1
vote
0answers
54 views

Find eigenfunctions of the integral operator with kernel $\sum\limits_{n=1}^\infty \frac{1}{n^2} \sin((n+1)x)\sin(ny)$

Find the eigenvalue and eigenfunctions of the integral operator $Ku=\int_0^\pi k(x,y)u(y)dy$. $k( x,y) = \sum\limits_{n=1}^\infty \frac{1}{n^2} \sin\big((n+1)x\big)\sin(ny)$. This is how I ...
1
vote
0answers
43 views

Is the Hankel Transform a Hankel Operator

The "Hankel Transform" is the infinite weighted sum of the Bessel function. At the top of the wikipedia article http://en.wikipedia.org/wiki/Hankel_transform it says Not to be confused with the ...
1
vote
0answers
16 views

Linear operator differentiation on a torus

I'm trying to analyze this article about area-preserving diffeomorphisms and don't quite understand a sentence. 4.1. Linear involutions. We start characterizing the linear involutions $R \! : ...
1
vote
0answers
32 views

Hamiltonian: Commutator

Given a Hilbert space $\mathcal{H}$. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ and an operator: $$A:\mathcal{D}(A)\to\mathcal{H}:\quad A=A^*$$ Denote for shorthand: ...
1
vote
0answers
37 views

weak closure of unitary group in $B(H)$

Let $H$ be a Hilbert space with dim $H=\infty$ , and $\cal{U}$ be the group of all unitaries on $H$. Show that the weak closure of $\cal{U}$ is a semigroup with respect to the multiplication. I know ...
1
vote
0answers
53 views

Explicit inverse of $\lambda-U$ when $U$ is unitary and $|\lambda|<1$

Let $U$ be a unitary operator on a Hilbert space $\mathcal{H}$. By the spectral theorem, it is known that $\sigma(U)\subseteq \{z\in \mathbb{C}:|z|=1\}$. How can the explicit inverse of $\lambda-U$ be ...
1
vote
0answers
55 views

Reducing subspaces of a normal operator

If $A$ is a normal operator on an infinite dimensional Hilbert space $H$, then $H$ is the direct sum of a countably infinite collection of subspaces that reduce $A$, all with the same infinite ...