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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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1answer
32 views

Find $\operatorname{ind}T$ where $T=\frac{d^2}{dx^2}$

Let $T$ be a linear and continuous operator defined as $$T=\frac{d^2}{dx^2}$$ Determine $\dim \ker T$ and $\dim \operatorname{coker}T$ in this two cases: $T: \mathscr{C}^2([a,b]) \longrightarrow \...
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1answer
25 views

Polar decomposition has unitary operator implies kernel of adjoint is trivial.

Let $T\in B(H)$ for $H$ a Hilbert space. Let $T = U|T|$ be the polar decomposition of a bounded operator $T$. I was able to prove that $U$ is unitary implies $\operatorname{ker}T = \{0\}$. However, $...
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1answer
54 views

How to prove this for the spectrum of a self-adjoint operator?

Let $T$ be a bounded self-adjoint operator. Prove: A number $\lambda \in \mathbb{R}$ belongs to the spectrum of $T$ if and only if $\mathbb{1}_{(\lambda - \epsilon, \lambda + \epsilon)} (T)$ is non-...
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1answer
45 views

What allows us to use the Heaviside operator like a variable?

We were taught to use the Heaviside operator $D: \dfrac{d}{dx}$ to solve an ODE, for example, Consider $y'' + 3y' +2y = e^{-2x}$ $$\implies (D^2 + 3D + 2)y = e^{-2x}$$ $$\implies y = \dfrac{1}{D^2 + ...
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1answer
18 views

Upper and Lower bounds on Normal Operator Norm

The problem comes from Naylor's Linear Operator Theory Section 5.23 Problem 14 Let $T$ be a normal operator on Hilbert space $H$ and let $T = A + iB$ be the Cartesian decomposition of $T$. Show ...
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1answer
31 views

spectrum of elements in $C^*$ algebra

Suppose $x,y$ are two invertible positive elements in a $C^*$ algebra $A$,if $\|x\|=\|y\|$,can we compute the spectrum $\sigma(x^{-1}y)$ of $x^{-1}y$?Does there exist a relationship between the ...
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17 views

Smallest eigenvalue of sum of two unbounded operators

Suppose $T,S:D(\mathcal H)\to \mathcal H$ are two unbounded operators with discrete spectrum consisting eigenvalues $0<\lambda_1(T)\leq\lambda_2(T)\leq\dots$ and $0<\lambda_1(S)\leq\lambda_2(...
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1answer
21 views

$\omega(A)=\|A\|\Longleftrightarrow C_A\cap \overline{W(A)}\neq \varnothing $?

Let $\mathcal{B}(F)$ the algebra of all bounded linear operators on a complex Hilbert space $F$. Let $A\in\mathcal{B}(F)$ and consider $$C_A:=\{z\in \mathbb{C}:\;|z|=\|A\| \}.$$ I want to prove ...
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1answer
24 views

comparision of two rank one operators

Suppose $S,T$ are two rank one operators on some Hilbert space $H$,If $S\leq T$,can we conclude that $T=kS$ for some $k\in \Bbb C$?
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1answer
17 views

compact operators are quasitriangular

A bounded operator $T$ on a Hilbert space is called quasitriangular if there exists an increasing sequence of finite rank projections $P_n$ converging pointwise to the identity such that $||TP_n-...
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1answer
57 views

Existence of Polynomial P such that P(A)=0

Let E be a Banach space of infinite dimension and A a compact operator of E in itself. I'm looking for this equivalence: There exists a polynomial P such that P (A) = 0 $\iff$ there exists $m \in \...
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1answer
20 views

Examples of continuous nonlinear operators in PDE

Let $X$ and $Y$ be Banach spaces and $D$ is a compact set in $X$. One can view a PDE as an operator and the solution should fall in the null space of it. Let $$ \mathcal{L}:X \mapsto Y, $$ where $\...
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1answer
11 views

Trace norm of partial trace of unitary transformation

Let $\mathcal{H}_{AB}$ be a bipartite, complex Euclidean space, and let $U\colon\mathcal{H}_{AB}\to\mathcal{H}_{AB}$ be a unitary operator. Define the trace norm as $$ \lVert X\rVert_1 = \text{Tr}(\...
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11 views

Norm of adjoint of operator of $l_1$

Let be $ T: l_{1} -> l_{1} $, $T(x_1,x_2,x_3,...)=(0,0,0,...x_1,0,0,...)$ . ( Coordinate x_1 is on n-place ). I proved that T is linear and bounded. But, I don't know how can prove that T is ...
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19 views

Is there a Deterministic Matrix with Restricted Isometry Property?

The Restricted Isometry Property (Low-Rank Matrices) Let $\mathcal{A}:\mathbb{R}^{n\times n}\rightarrow \mathbb{R}^m$ be a linear operator. The constant $\delta_r:=\delta_r(\mathcal{A})$ is defined ...
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1answer
41 views

Example of a strongly dense set which is not ultrastrongly dense.

Let $H$ be a Hilbert space and let $B(H)$ denote the Banach space of bounded operators on $H$. Then there are several topologies we can endow on $B(H)$. I am interested in the case of the strong ...
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87 views
+50

Explicit solution of an operator equation

Suppose we have an operator $A:C([0,\infty )) \to C([0,\infty ))$ defined as $$(Af)(x)=\int _0 ^ x \dfrac{f(y)}{\sqrt{x^2-y^2}}dy$$ Now I want to prove that for arbitrary $g\in C([0,\infty ))$ the ...
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25 views

existence of eigenfunctions in non-self adjoint operator

Remark: The existence of eigenfunctions is only guaranteed for self-adjoint operators. How does one justify this statement? An operator not being self-adjoint makes it not have eigenpairs?
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22 views

How do we define the norm of a block operator matrix?

Consider the block-operator matrix $$ D = \begin{bmatrix} D^{1,1} & D^{1,2} & \dots & D^{1,M} \\ D^{2,1} & D^{2,2} & \dots & D^{2,M} \\ \vdots & \vdots & \ddots & \...
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26 views

Capacity limitations

Just to ensure that what's the current day status of the computational capacity and what are it's limitations till date, on several mathematical operations listed below - 1.)Suppose $n$ is any ...
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1answer
61 views

Showing that $\lambda$ is an eigenvalue iff $\mathbb{1}_{\left\{\lambda\right\}} (T)$ is non-zero

Let $T$ be a bounded self-adjoint operator. I already showed that the operator $\mathbb{1}_{U} (T)$ is an orthogonal projection, where $\mathbb{1}_{U}$ is the characteristic function of the Borel set $...
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1answer
49 views

Construct a vector so the operator inequality $\|Av\|_1 \le \|A\|_1 \|v\|_1$ is equal

First, we should show that for $A\in \mathbb{R}^{m\times n}$: $\|Av\|_1 \le \|A\|_1 \|v\|_1$. $$(1)\quad\|Av||_1=\sum_{i=1}^m|(Av)_i| \le \sum_{i=1}^m \sum_{j=1}^n|A_{i j}||v_j|=\sum_{j=1}^n (\sum_{...
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1answer
30 views

Why $\lim_{n\rightarrow\infty} \mbox{Re}(T_1\xi_n\mid \lambda T_2\xi_n)=\lim_{n\rightarrow\infty} (T_1\xi_n\mid \lambda T_2\xi_n)$?

Let $\mathbb{B}(\mathcal{H})$ denote the $C^*$-algebra of all bounded linear operators on a complex Hilbert space $\mathcal{H}$ endowed with an inner product $(.\mid.)$. Let $T_1,T_2\in\mathbb{B}(\...
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13 views

Prove that finite difference operator is commutative

I'm trying to prove the symmetry of mixed partial derivatives in which the following lemma is appealed. $\textbf{Lemma:}$ Let $X$ be open in $\mathbb R^n$, $f:X \to F$, and $m \in \mathbb N$. We $h ...
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1answer
19 views

Class of Fourier multiplier in $L^1$ is the class of Fourier transform of finite Borel measures. (Stein)

Hi. I am trying to prove an observation in Stein, singular integrals. Observation. $\mathcal{M}_{1}$ (class of Fourier multiplier in $L^1$) is the class of Fourier trasnforms of elements of $\mathcal{...
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14 views

to show unboundedness of an operator

Given $l$ is an even, $2\pi-$periodic, $L^2$ function. An operator $L: H \to H$, where $H$ is a Hilbert space, is defined as $$Lf:=\int_{-\pi}^\pi l(x-y)f(y)dy.$$ To show that $L$ does not have ...
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1answer
25 views

Find the operator P(x,y,z) [closed]

I need some help with this problem. $P:\mathbb{R}^3\to\mathbb{R}^3$ the linear operator such that $u=P(v)$ is the orthogonal projection of v E $\mathbb{R}^3$ on plane $3x+2y+z=0$. Find $P(x,y,z)$ (...
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20 views

Does convergence for increasing sequences of projections imply convergence for all sequences of projections?

Let be $\{P_n\}_n$ an increasing sequence ($P_{n+1} \ge P_n$) of finite dimensional projection operators on a separable Hilbert space $H$. Let $f:H \to \mathbb{C}$. Let us fix $f_n(x):=f(P_nx)$. If $...
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0answers
13 views

Eigenvalue of Hermitian operator(curl) complex?

This question is in continuation to a previous question What are the Eigenvectors of the curl operator?. I know now that the curl operator \begin{bmatrix} 0 & -\frac{\partial}{\partial z} &...
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23 views

Passive transformation by antiunitary operator and orientation of the complex plane.

I have recently tried to make sense of the concept on an antiunitary passive transformation on a complex Hilbert space $H = \mathbb{C}^N$. I still do not know whether the concept even makes sense ...
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58 views

Showing that $U$ is unitary iff this kernel contains only zero vector

In our introductory functional analysis class, we proved the following: Theorem: Let $T \in B(H)$. Denote by $P$ the orthogonal projection onto $(\ker(T))^{\perp}$. There exists a unique operator $U \...
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0answers
24 views

Null and Range Projections

First some notation/terminology. Let $T$ be some bounded linear operator acting on a Hilbert space $\mathcal{H}$. The null space is $\{x \in \mathcal{H} \mid Tx=0\}$ and the range space is the (norm?) ...
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19 views

Regularity of eigenfunctions of an ordinary differential operator

Let $I \subseteq \mathbb{R}$ be an open interval and $T:D(T)\subseteq L^2(I) \to L^2(I)$ a differential operator given by $$(Tf)(x):= \sum_{j=0}^n a_j(x)f^{(j)}(x), \quad f\in D(T), \ x \in I,$$ where ...
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29 views

Adjoint of this operator?

Problem: Let $\lambda \in l^{\infty} (\mathbb{Z})$. Define the operator $$ T: l^2 (\mathbb{Z}) \rightarrow l^2 (\mathbb{Z}): (Tx)(n) = \lambda(n) x(n+1). $$ What is $T^{*}$? Attempt: The adjoint ...
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1answer
58 views

Vector field where curl of the field is the field itself

I was curious if there is a possibility of a field where the curl of the field would be the field itself, i.e, $$\nabla \times \vec{A} = \vec{A}$$ I can immediately see that the divergence of such a ...
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1answer
15 views

Relation between spectrum and operator support

$f$ is a function in $C^{\infty}$ with compact support and $A$ is a self-adjoint operator. If $\mathrm{supp}(f)\cap\sigma(A)=\varnothing$, does this imply that $f(A)=0$? Does anyone have an ...
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1answer
40 views

Please Help with Question about Linear Operators

The Question: Let $X$ and $Y$ be Banach spaces and $T: X \rightarrow Y$ an injective bounded linear operator. Show that if $R(T)$ is closed in Y, then $T^{-1} : R(T) \rightarrow X $ is bounded. My ...
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15 views

Abelian/Finite Projections in Von Neumann Algebras

I know that provided a von Neumann Algebra acting on Hilbert $(\mathcal{M},\mathcal{H})$ a projection $e \in P(\mathcal{M})$ is said to be abelian if $e\mathcal{M} e$ is $\textbf{abelian}$ and the ...
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2answers
26 views

On polar decomposition

In the paper "Isometries of non-commutative $L_p$-spaces" by Yeadon the author states the following: Let $H$ is separable Hilbert space, $\mathcal B(H)$ is $\ast-$algebra of all bounded linear ...
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14 views

Bound for the norm of a symmeric Markov kernel acting on $L^2$ in terms of the spectral gap

Let $\kappa$ be a symmetric Markov kernel on a probability space $(E,\mathcal E,\mu)$. We know that $$\kappa_0 f:=\int\kappa(\;\cdot\;,{\rm d}y)f(y)\;\;\;\text{for }f\in L^2_0(\mu)$$ is a self-adjoint ...
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18 views

If $A$ is self-adjoint, does the formula $r(A)=\max_{\lambda\in\sigma(A)}|\lambda|$ for the spectral radius even hold in the non-complex setting?

Let $\mathbb K\in\left\{\mathbb C,\mathbb R\right\}$, $H$ be a $\mathbb K$-Hilbert space, $A\in\mathfrak L(H)$ and $\sigma(A)$ and $r(A)$ denote the spectrum and spectral radius of $A$, respectively. ...
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0answers
35 views

If $A$ is self-adjoint, then $\left\|A\right\|=\max_{\lambda\in\sigma(A)}|\lambda|$

Let $H$ be a $\mathbb R$-Hilbert space, $A\in\mathfrak L(H)$ be self-adjoint with $\left\|A\right\|_{\mathfrak L(H)}\le1$, $\sigma(A)$ and $r(A)$ denote the spectrum and spectral radius of $A$, ...
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1answer
47 views

Studying the off-diagonal case of this operator $f \mapsto \int_{B(0,1)} f(x-y)dy $

I am trying to show that the operator $T$ defined on Schwarz class functions (on $\Bbb R^n$) by, $$f \mapsto \int_{B(0,1)} f(x-y)dy $$ (where $B(0,1)=\{t \in \Bbb R^n : ||t|| <1\}$ ) , is not ...
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20 views

Book on Operator Theory

I am new to things like the Contraction Mapping and am looking for a book to further my understanding. Are there any recommendations? Also, I wonder if one needs to have a good understanding of ...
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1answer
24 views

$TT^*$ is unitary equivalent to $T^*T$.

Let $H$ be a complex Hilbert space and let $\mathcal{B}(H)$ denote the bounded linear operators from $H$ to itself. An operator is said to be normal if $TT^* = T^*T$. I would like to know which ...
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0answers
18 views

Operator norm of symmetric Markov kernel acting on $L^2$

Let $(S,\mathcal B)$ be a measurable space, $\pi$ be a Markov kernel on $(S,\mathcal B)$ and $\mu$ be a probability measure on $(S,\mathcal B)$ invariant with respect to $\pi$. Let $L^2_0(\mu):=\left\{...
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1answer
58 views

If $A$ is self-adjoint, then $\left\|A\right\|=\sup_{x\in H\setminus\{0\}}\frac{\langle Ax,x\rangle}{\left\|x\right\|^2}$

Let $H$ be a $\mathbb R$-Hilbert space and $A\in\mathfrak L(H)$ be self-adjoint. I want to show that $$\left\|A\right\|_{\mathfrak L(H)}=\sup_{x\in H\setminus\{0\}}\frac{\langle Ax,x\rangle_H}{\left\|...
3
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0answers
44 views

Linear Operator in a Banach Space

I'm trying to get my head around linear operators and their usage with Banach spaces. Could someone help me understand how some properties relate to the following operator? We have Banach space $L^2$,...
2
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0answers
30 views

Spectrum of compact operator on an infinite dimensional Banach space

Let $X$ be an infinite dimensional Banach space (over $\mathbb{C}$) and let $T\colon X\to X$ be a compact operator. Let $\sigma(T)$ denote the spectrum of $T$ and let $\sigma_{\text{p}}(T)$ denote the ...
1
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0answers
28 views

comparision with rank one projection

Suppose $P$ is a rank one projection in $B(H)$,if we have $T\in B(H)$ such that $0\leq T\leq P$,does the following conclusion hold: There exists $\alpha \geq 0$ such that $T=\alpha P.$