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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2
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63 views

Show compactness of an evolution operator

Consider the heat equation $$ u_{t}=u_{xx},~~~~~u_0(x)=u(0,x)$$ with $u\colon [0,T]\times\mathbb{R}\to\mathbb{R}, (t,x)\mapsto u(t,x)$ and the evolution operator $E(T)$ with $E(T)u_0=u(T,x)$. 1.) ...
2
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1answer
270 views

A generalization of the Cauchy-Schwarz inequality to linear operators

If $A$ is an operator and $A \in \mathcal{B_{+}(X)}$ (the set of the positive operators) then the generalization of the Cauchy-Buniakowsky-Schwarz inequality holds: $$|\langle Ax,y\rangle| \leq ...
2
votes
1answer
91 views

Why is $\langle Ax, Ax \rangle = \langle A^2 x, x\rangle$?

Let $X$ be a Hilbert space and $A\in \mathcal{B}(X)$ be self-adjoint. How can I prove: $$\langle Ax, Ax \rangle = \langle A^2 x, x \rangle$$ I know it is a simple problem, but I don't know how to ...
2
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0answers
241 views

Determining the spectral representation of a operator

The spectral representation for a self-adjoint operator $T \in L(H)$ for H a Hilbert space is written as: $$ T = \sum_{\lambda \in \sigma(T)} \lambda \pi_{\lambda}, $$ where $\sigma(T)$ is the ...
1
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0answers
29 views

How to show that density?

Show $$ \overline{\operatorname{span}(v_j)}=L^2([0,1]),~~~~~\overline{\operatorname{span}(u_j)}=L^2([0,1]) $$ with $$ v_j(x)=\sqrt{2}\cos((j-1/2)\pi x),~~~~~u_j(x)=\sqrt{2}\sin((j-1/2)\pi x). $$ ...
3
votes
1answer
65 views

Spectrum of operator on canonical orthonormal system

Define the operator $T: l^2 \rightarrow l^2$ on the canonical orthonormal system $(e_k)_k$ by: $$ Te_k := \frac{e_k}{k} + \frac{e_{k+1}}{k+1}, $$ such that for $a\in l^2$: $$ T((a_i)_i) = (a_1, ...
2
votes
0answers
79 views

Verify a given SVD of an operator

Show that the Singular Value Decomposition of the operator $$ A\colon L^2([0,1])\to L^2([0,1]), x\mapsto\int\limits_0^t x(s)\, ds $$ is given by $$ ...
4
votes
2answers
100 views

If $A\leq B$ in the sense of quadratic forms, then must $A^{-1} \geq B^{-1}$?

Let $A$ and $B$ be symmetric invertible operators on a Hilbert space $X$. Suppose $$ \langle Ax , x \rangle \leq \langle Bx , x \rangle $$ for each $x\in X$. Does it follow that $\langle A^{-1} x ,x ...
3
votes
0answers
61 views

A set of trajectories as a linear subspace of Hilbert space

Let $\left\{S(t)\right\}_{0 \leqslant t \leqslant \theta}$ be a strongly continuous semigroup of linear continuous operators in Hilbert space $H$, $S(0) = I$. Let $x$ be some element of $H$. Then its ...
2
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1answer
91 views

Evolution operator

We call a function that assigns a starting value of a time-dependent differentialfunction to a solution of a later timevalue as the evolution operator $E(t)$. Look at the thermal equation $$ ...
3
votes
1answer
86 views

Singular Value Decomposition - what do I have to do?

Show that the Singular Value Decomposition of $$ T\colon L^2([0,1])\to H^1([0,1]), x\mapsto\int\limits_0^t x(s)\, ds $$ is given by $$ \sigma_j=\frac{1}{(j-1/2)\pi}, v_j(x)=\sqrt{2}\cos((j-1/2)\pi ...
3
votes
2answers
232 views

Find adjoint operator of an operator T

I would like to find the adjoint operator of $$ T\colon L^2([0,1])\to H^1([0,1]),\quad x\mapsto\int\limits_0^t x(s)\, ds. $$ Here $H^1([0,1])$ is the Sobolev space $W^{1,2}([0,1])$. I tried to find ...
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votes
1answer
106 views

If a,b ‎are ‎unitary ‎equivalent,‎Dose ‎ ‎‎$‎\sigma(a)=‎\sigma(b)‎$‎ is true?

‎‎Let A‎ ‎is ‎an ‎unital‎‎ ‎algebra ‎and ‎‎$ ‎Ad‎~u:‎‎‎A\rightarrow ‎A~,~a‎\mapsto~‎uau‎^{*}‎‎$ ‎and u‎ ‎is ‎unitary ‎element ‎of A‎(‎$‎uu‎^{‎*‎}=‎u‎‎^{*}‎u=1‎$‎), ‎if ‎‎$‎b=‎uau‎^{‎*‎}‎‎$ ‎(a,b ‎are ...
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1answer
140 views

Compactness and spectrum of an intergral operator.

Let $u \in L^\infty ([0,1])$ be fixed and define the operator $T$ on $C([0,1])$ by $$ Tf(x) = \int_0^x u(t) f(t) dt $$ Show that T is compact and into itself and determine its spectrum. My try: The ...
4
votes
1answer
76 views

Nullspace of $T'$ for continuous bounded functions.

Given $f\in C_b(\mathbb{R})$, let $Tf(x) = e^{-|x|}f(x)$. Show that $T$ defines a bounded linear map into itself such that $\ker T' \neq {0}$. My try: look at the space $A = \{ f \in C_b : ...
14
votes
1answer
200 views

$A$ and $B$ commute on a dense set but $e^{iA}$ and $e^{iB}$ do not

Let $A$ and $B$ be unbounded, symmetric operators on a Hilbert space $H$ with a common domain $D$. If $AB = BA$ on $D$, is it necessarily that case that $e^{iA}$ and $e^{iB}$ also commute? If $A$ and ...
4
votes
2answers
176 views

Functional analysis-Closed graph therem

Let $ X$, $ Y$, $ Z$, be Banach spaces and let $ T:X\to Y $ and $ S:Y\to Z $ be linear transformations.Suppose that $S$ is Bounded and injective and that $ S \circ T $ is bounded.Prove that $T $ is ...
4
votes
1answer
585 views

When to use Closed Graph Theorem vs. Uniform Boundedness Theorem?

I run in to problem that I often know is solvable with either the Closed Graph Theorem or Uniform Boundedness Theorem. I seem to mix up the solutions. Are there any hints on when to use which? Or can ...
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votes
1answer
133 views

Functional analysis - bounded linear transformation

Let $ \mathcal{H} $ be a Hilbert space, and let $ T: \mathcal{H} \to \mathcal{H} $ be such that $ \langle x,Ty \rangle = \langle Tx,y \rangle $ for all $ x,y \in \mathcal{H} $. How can one show that ...
2
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244 views

definition of operator valued integral with spectral measure

I am trying to make sense of some operators that come up on Buchholz and Summers' work on warped convolutions (two works on arxiv: 2008 and 2011). There, they work on a Hilbert space $H$ and on the ...
2
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0answers
129 views

winding number question

This is part of a proof from Banach algebra techniques in Operator theory by Ronald Douglas on page 170. Let $\epsilon>0$. Let $T$ be the unit circle and $\phi\in H^\infty+C(T)$. Choose $\psi\in ...
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votes
0answers
38 views

Normal operators on separable Hilbert space [duplicate]

Possible Duplicate: Normal operators in Hilbert spaces Let $H$ be a separable Hilbert space and let $T \colon T \rightarrow T$ be a continuous linear map such that there exists an ...
3
votes
1answer
482 views

Compactness and spectrum of integral operator

Show that the operator $C: L^2([0,1]) \rightarrow L^2([0,1])$ defined by $$Cf(x) = \int_0^x\int_1^tf(s)dsdt$$ is compact and determine its spectrum. Im not sure how to find the spectrum when we are ...
3
votes
1answer
175 views

Bounded operators on separable Hilbert spaces

Let $H$ be a separable Hilbert space. Show that every bounded operator from $H$ to itself can be approximated in the strong operator topology by a sequence of finite rank operators. Im not sure what ...
6
votes
1answer
286 views

Trace class for operators

Let $ \mathcal{H} $ be a Hilbert space and $ T: \mathcal{H} \to \mathcal{H} $ a bounded linear operator. The $ n $-th singular number $ {\mu_{n}}(T) $ of $ T $ is defined as the distance from $ T $ ...
10
votes
1answer
678 views

Importance of Toeplitz operators?

I am reading Arveson's A Short Course on Spectral Theory, in which the author states that Toeplitz operators are very important without giving references on their applications. After some searching, I ...
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84 views

eigenvalue question for a Toeplitz operator

Let $\phi$ be a nonzero function in $L^\infty(T)$ where $T$ is the unit circle. Let $M_\phi$ be the multiplication operator and $T_\phi$ be the Toeplitz operator. Show $T_\phi$ and $M_\phi$ have no ...
8
votes
2answers
233 views

Compact maps problem in Lax

In Functional Analysis of Peter Lax there are the following exercise Show that if $\bf C$ is compact and $\{{\bf M}_n \}$ tends strongly to $\bf M$, then $\bf CM_n$ tends uniformly to $\bf CM$. ...
7
votes
1answer
717 views

$T$ surjective iff $T^*$ injective in infinite-dimensional Hilbert space?

Let $T:H_1\rightarrow H_2$ be a bounded linear operator where $H_1$ and $H_2$ are Hilbert spaces. The Hilbert-adjoint is defined to the the operator $T^*:H_2\rightarrow H_1$ such that $\langle ...
5
votes
2answers
469 views

Norm inequality for sum and difference of positive-definite matrices

If $X_{1}$ and $X_{2}$ are positive definite matrices, how to show that $\left\Vert X_{1}-X_{2}\right\Vert \le\left\Vert X_{1}+X_{2}\right\Vert$ for the spectral norm? and how about for the nuclear ...
6
votes
1answer
132 views

invariant subspace of a Hardy space

Let $T$ be the unit circle and $H^1=\{f\in L^1(T): \int_0^{2\pi} f(e^{it})\chi_n(e^{it})dt=0 \text{ for } n>0\}$ where $\chi_n(e^{it})=e^{int}$. Let $M$ be a closed subspace of $H^1$. Then ...
5
votes
1answer
324 views

Fourier transform as a Gelfand transform

One question came to my mind while looking at the proof of Gelfand-Naimark theorem. Is Fourier transform a kind of Gelfand transform? Are there any other well-known transforms which are so?
10
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2answers
974 views

Gelfand-Naimark Theorem

The Gelfand–Naimark Theorem states that an arbitrary C*-algebra $ A $ is isometrically *-isomorphic to a C*-algebra of bounded operators on a Hilbert space. There is another version, which states that ...
4
votes
0answers
79 views

The control of norm in quotient algebra

Let $B_1,B_2$ be two Banach spaces and $L(B_i,B_j),K(B_i,B_j)(i,j=1,2)$ spaces of bounded and compact linear operator between them respectively. If $T \in L(B_1,B_1)$, we have a $S \in K(B_1,B_2)$ and ...
2
votes
2answers
718 views

Weak limit and strong limit

Let $X$ be a Banach space and let $x_n \overbrace{\rightarrow}^w x$ and $x_n \overbrace{\rightarrow}^s z$ can we then say that $x = z$? My try: $$\| x- z\| = \sup_{\ell \leq 1} |\ell(x-z)| = ...
6
votes
1answer
97 views

local convexity of $L_p$ spaces

wiki says The spaces $L_p([0, 1])$ for $0 < p < 1$ are equipped with the F-norm they are not locally convex, since the only convex neighborhood of zero is the whole space Why is this so? ...
2
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1answer
102 views

Normal Operator surjective

I have difficulty proving: If $T$ is a normal operator in a Hilbert space, $T$ is surjective if and only if $T^*$ surjective. Please give me some help. Thank you.
4
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1answer
137 views

Multiplication operator and trace class

Suppose we work in $H=l^2(\Bbb{N})$ and suppose the multiplication operator $T_f$ such that $T_f\psi=f\psi$ and $f:\Bbb{N}\rightarrow \Bbb{C}$. We denote by $B_1(H)$ the trace class of operators. ...
2
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1answer
65 views

Uniqueness of for integration functional

Let $f\in C([0,1])$ and assume that there exists a positive constant C such that $\left| \int_0^1p'(t)f(t) dt \right| \leq C\|p\|_2 $ for all polynomials $p$, where $\|p\|^2_2 = \int_0^1 |p(t)|^2 dt$. ...
1
vote
1answer
120 views

Continuous maps in from Banach space to $\ell ^\infty$

Let $X$ be a Banach space. Prove that a linear map $M\colon X\mapsto \ell^p, \; p\geqslant 1$ is continuous iff for every sequence $(x_k)$ that converges in $X$ to $x \in X$, we have that the $n$-th ...
9
votes
1answer
172 views

Do we have Maximal Abelian Algebras (MAAs)?

Let $\mathcal{H}$ be a Hilbert space and $B(\mathcal{H})$ the algebra of bounded linear operators on $\mathcal{H}$. A MASA $\mathcal{M}$ is a subalgebra of $B(\mathcal{H})$ that is abelian and ...
2
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0answers
93 views

Inverse of a certain differential operator (resolvent)

I am doing a research on a certain type of operator, and in the course of it I need to determine the following: Given the operator $D$ below, and identity operator $I$, $$ D=\begin{pmatrix} ...
6
votes
1answer
96 views

Uniform limit of finite-rank operators with the same rank.

Let $\{T_n\in\mathcal{B}(X)\,|\,\text{rank}(T_n)=R\,\}^{\infty}_{n=1}$ is a sequence of linear bounded finite-rank operators on a Banach space with the same rank $R$. Let it converge uniformly to an ...
3
votes
2answers
91 views

Compactness of operator $M: C([0,1]) \rightarrow C([0,1])$

Let $M: C([0,1]) \rightarrow C([0,1])$ be defined by $$ Mf(x) = f(x/2), \;\; x\in[0,1]$$ Is this operator compact? I have trouble using limit in operator norm of compact operator, or cauchy ...
3
votes
1answer
79 views

Range of operator $ Mf(x) = f(x/2), \;\; x\in[0,1]$

Let $M: C([0,1]) \rightarrow C([0,1])$ be defined by $$ Mf(x) = f(x/2), \;\; x\in[0,1]$$ Prove that the range of $I-M$ does not contain nonzero constant functions, but it contains all functions ...
4
votes
1answer
107 views

bounded operator between continuous functions

Let $M: C([0,1]) \rightarrow C([0,1])$ be defined by $$ Mf(x) = f(x/2), \;\; x\in[0,1]$$ Show that $M$ is bounded and that its spectrum is containd in the closed unit disc $\{ \lambda \in \mathbb{C} ...
0
votes
1answer
60 views

Subspace in $I-T$ for bounded linear maps

Let $X$ be a Banach space and let $T:X \rightarrow X$ be a bounded linear map.Show that the range of $I - T$ contains the subspace $$Y_T = \{x \in X: \limsup_{n\rightarrow \infty} n^2\|T^nx\| < ...
4
votes
1answer
255 views

Bounded linear maps in Banach spaces

Let $X$ be a Banach space and let $M: X \rightarrow X$ be a linear map. Prov that M is bounded iff there exists a set $S \subset X'$, dense in X', such that for each $\ell \in S$ the functional $m_l$ ...
2
votes
1answer
91 views

Closed extensions in the weak* topology

Let $\ell^\infty$ be the Banach space of bounded sequences with the usual norm. and let $\ell_0(x) = \lim_{n \rightarrow \infty} x_n$, for convergent sequences. Show that the sett L consisting of all ...
3
votes
2answers
462 views

Show that a finite-dimensional Banach space has a bijective compact operator

It is clear that if $ T: X \rightarrow X $ is a bijective compact operator, where $ X $ is a Banach space, then $ \dim(\text{Range}(T)) = \dim(X) $, which implies that $ \dim(X) $ must be $ < ...