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
1answer
37 views

Operator norm: Show that $\|A\|=\sup\{ |f(Ax)| : x \in X, \|x\| \leq 1 , f \in Y^*, \|f\| \leq 1 \}$

Good day, As stated in the title, I have to show that $\|A\|=\sup\{ |f(Ax)| : x \in X, \|x\|_X \leq 1 , f \in Y^*, \|f\| \leq 1 \}$ where $\| \cdot \|$ is the operator norm, i.e. for $X,Y$ vector ...
0
votes
0answers
30 views

Ladder (recurrence) operators for Hermite polynomials?

Generally, Hermite polynomials can be described using the Rodrigues formula: $$ H_n(x) = (-1)^n e^{x^2}\frac{d^n}{dx^n}e^{-x^2} $$ And the first few polynomials (for $n = 0,1,2,3,4,5...$) are well ...
0
votes
1answer
39 views

Adjoint of a matrix vs adjoint operator

I am having some confusion over how I would attack a proof of the properties of matrix adjoints. Here is an example: Let $A,B\in M_{n\times n}(\mathbb{F})$ with adjoints $A^*$ and $B^*$. Prove ...
0
votes
1answer
47 views

Problem involving the Spectral Mapping theorem.

Consider the following problem: Let $T$ be a bounded operator in a Banach space $X$. Use the Spectral Mapping theorem to show that $|\lambda^n|\le\|T^n\|$ for all $\lambda\in\sigma(T).$ Here's ...
1
vote
2answers
59 views

Difference between the spectrum and point spectrum of an operator.

I have the following two definitions in my notes: The spectrum of an operator: We define $\sigma(T)$, the spectrum of T, by, $$\sigma(T):=\{\lambda\in\mathbb C: T-\lambda I\,\, \text{is not ...
0
votes
0answers
15 views

How do we take the limit of this quantum operation?

I am wondering how to take the following limit: \begin{align} L= \lim_{\tau \to \infty} \frac{1}{\tau} \int_{-\tau/2}^{\tau/2} dy \, \left(1 - \frac{1}{\sqrt{ \pi} \sigma } ...
0
votes
0answers
32 views

Norm of direct sum of operators acting on complemented subspaces of a Banach space.

Suppose $X$ is a Banach space with a normalized Schauder basis $\{e_k\}$ and basis constant $1$. This means for each $x$ in $X$ there is a unique sequence of scalars $\{a_n\}$ for which ...
0
votes
1answer
28 views

A lower bound on the form of the resolvent operator

Let $A\in\mathbb{C}^{n,n}$ and $x\in\mathbb{C}^{n}$, $\|x\|=1$. Is there any $c(z)>0$ such that $$|\langle x, (A-z)^{-1} x \rangle|\geq c(z), \quad \text{ for } |z|>\|A\|\,?$$ Recall that it is ...
1
vote
1answer
33 views

Construct a projection satisfying a certain property

Let $\cal G$ be a group of finite order $n$. For every prime divisor $p$ of $n$, construct a projection $P\in \cal N(G)$ such that $\operatorname{tr}_{\cal N(G)}(P)=1/p$. Here $\cal N(G)$ denotes ...
3
votes
1answer
43 views

Determine all $T \in B(C[0,1])$ that commute with the multiplication operator $Mf=xf(x)$

Let $Mf=xf(x)$ and for any polynomial let $T$ be a "polynomial operator" $Tf=(a_{x}x^{n}+..+a_{0})f$ Clearly $M$ commutes with all such "polynomial operators" and therefore since the weak limit of ...
3
votes
1answer
30 views

Given diagonal Matrix and number $q$, find nilpotent matrix such that $D N D^{-1} = qN$.

Given a diagonal matrix $D$ and an integer $q$, find a nilpotent matrix $N$ such that $$DND^{-1}= qN$$ holds. Just thinking about this for a bit, I'm not convinced it's possible, since the ...
0
votes
1answer
28 views

How to construct a non-trivial example that shows $A^{-1}A = I_X $ and $AA^{-1} = I_Y$, where $I_X \neq I_Y$

Let $A: X \to Y$, and $A^{-1}:Y\to X$ be maps where $A^{-1}$ is the inverse of $A$ Then $A^{-1}A = I_X$ and $AA^{-1} = I_Y$ I can't think of good examples to demonstrate this result, because for ...
1
vote
1answer
29 views

How to find spectrum of a convolution operator

Say $k$ be s.t. $\hat{k}$ is a bounded function on an LCA group $G$ and $Tf=f*k$. Then $T$ is bounded on $L^2(G)$. Is there anything I can say about $\sigma(T)$? (except the properties that follow ...
0
votes
0answers
48 views

The dual norm of a operator matrix norm

Let as look at matrices $B$ in $\mathbb{R}^{p\times q}$ together with the following operator norm: $$||B||_{op}:=\max_{\beta}\frac{|B\cdot \beta|_{p}}{|\beta|_{q}}.$$ Here $|\cdot|_{p}$ is any norm on ...
1
vote
1answer
35 views

Why an unbounded operator defined everywhere fails to be closed?

The Toeplitz theorem says : If a closed operator is defined everywhere, then it is continuous. So if a non continuous operator is defined everywhere, it is not closed. But why is it not closed? What ...
0
votes
0answers
25 views

Lumer-Phillips Theorem for non-contraction semigroups?

Let $H$ be a closed operator on a Hilbert space $\mathcal H$. The Lumer-Phillips theorem states that $H$ is a generator of a one-parameter contraction semigroup if and only if $\Re\langle ...
1
vote
1answer
31 views

Operator between Hilbert spaces, boundness, image and eigenvalues

I'm totally new in functional analysis and this is my first problem. Let's $H=L^2(-\pi,\pi)$ as Hilbert space with basis $u_n+iv_n$ where $$u_0 = ...
5
votes
2answers
95 views

Consider the Banach Space $C[0,1]$. Find decomposition of spectrum of the indefinite integral operator.

Cosider the Banach Space $C[0,1]$ of real-valued continuous function on $[0,1]$ with the supremum norm. and the linear operator $$A: x(t)\mapsto\int\limits_0^tx(s)ds$$ Find its eigenvalues, ...
1
vote
2answers
34 views

Existence of Unitary Map

I've been recently introduced to Unitary operators of a Hilbert space and I've been wondering the following. Existence of a unitary operator $T$ on a (possibly infinite) Hilbert space $H$ is simple ...
1
vote
1answer
59 views

Positive bounded operators

Let $A,B$ be positive self-adjoint bounded operators and $\lambda >0$ then I want to show that if $$A-B \ge 0 $$ that is $\langle x,(A-B)x \rangle \ge 0$ we have that the resolvents (whose ...
1
vote
1answer
32 views

The relation between Closed Operators (the graph is closed) and Closed Mappings (images of closed sets are closed)

Let $X$, and $Y$ be topological vector spaces and let $D$ be a dense vector subspace of $X$. An operator $T:D\to Y$ is called closed iff the graph of $T$, $\{(x,T(x))\in X\times Y|\,x\in D\}\subseteq ...
1
vote
0answers
29 views

Can we use a series of properties to determine integral operator $f \to \int_0^1 f d\mu $

Question: Suppose there exists an operator $I: C^{\infty}(0,1) \to \mathbb R$ satisfying the following properties: (1) $I (\chi_{(0,1)})=1$ ; (2) $I(kf)=kI(f)$, where $k\in \mathbb R$ and $f\in ...
0
votes
2answers
62 views

prove that all pure states in a commutative C* algebra are multiplicative linear functionals

I am trying to prove this , but can not see it clearly. it was given as some sort of converse of the fact that all multiplicative linear functionals are pure states
1
vote
1answer
29 views

Find basis for exact matrix form of linear operator

$A:\cal{P}_1 \to \cal{P}_1$ is a linear operator defined with $$ A(p)(t):=(3t+1)p'(t)+2p(t). $$ I'm trying to find a basis $e$ of $\cal{P}_1$ in which the operator $A$ has the matrix form $$ A= ...
4
votes
1answer
60 views

Find the eigenvalues of the operator T.

I have the following problem, "Suppose that $X=\ell^1$ and define the operator $T\in B(X)$ as follows: $$Tx=\left(\frac12x_2,\frac13x_3,\frac14x_4,...\right)\,,\textit{where,}\,\,\, ...
2
votes
1answer
33 views

Inverse continuity of an operator

Let $X$ be a Banach space (it is in fact an $L^p$ space) and let $T:X \to X$ be a linear continuous operator (which is not injective and not surjective). I am trying to figure out if the following is ...
4
votes
1answer
46 views

Fundamental solution of a shifted operator

what is the fundamental solution of the shifted operator $ \Delta + \lambda^2 $, i.e, what the function $f$ satisfying the following equation $$ (\Delta + \lambda^2 )f(x) = \delta(x),$$ where $ \Delta ...
2
votes
1answer
191 views

idempotents in a subalgebra of $B(H)$.

Let $\mathcal{A}$ be a sub-algebra of $B(H)$ such that $\mathcal{A}$ generated by all its idempotents and $\mathcal{A}$ is closed under weak operator topology. Suppose that there exist idempotents ...
1
vote
1answer
21 views

Forward difference operator

What does $\Delta^{-1}$ mean? I have seen it in a question such as "justify that $\Delta^{-1}k^{(n)} = {k^{n+1}\over{n+1}}$". Thanks for your help.
2
votes
1answer
30 views

$\omega$ is cyclic for $M\subset B(H)$ if and only if $\omega$ is separating for $M'$

Let $H$ be a Hilbert space, $M\subset B(H)$ a von Neumann algebra and $\omega \in H$ a vector. Then $\omega$ is cyclic for $M$ if and only if $\omega$ is separating for $M'$. I proved ...
3
votes
0answers
24 views

A problem on left Fredholm Operator..

I was reading Fredholm Operators from the book "A course in Functinal Analysis " by J.B Conway. There I got stuck with the following problem. Let $A\in B(\mathcal H)$. Show that $A(\mathcal M)$ is ...
0
votes
2answers
41 views

Calculating operator (matrix) norms using eigenvalues?

A remark that went unproven in class. It was said that the operator norm of a real linear transformation (real matrix) is the square root of the abs value of the max eigenvalue of $A^T*A$ (or maybe ...
0
votes
1answer
34 views

Left Shift Operator Spectrum Q2

Consider $\ell^2(\mathbb{Z})$. Let $R: \ell^2(\mathbb{Z}) \to \ell^2(\mathbb{Z})$ be such that $R((a_n)) = (a_{n+1})$. I need to prove that, given $z \in \mathbb{C}$ with $|z| >1$, the two series ...
1
vote
0answers
26 views

Riemann Lebesgue Lemma for locally compact ableian groups

I'm looking for a reference (or proof here) of the generalized RL Lemma for LCAGs. One result is that if $G$ is a LCAG then $$\{ \hat{f} : f \in L^1(G)\} \subset C_0(\Gamma)$$ where $\Gamma$ is the ...
4
votes
1answer
63 views

Ranges of projection operators

Suppose that $X$ is a Banach space and $P$ and $Q$ be bounded linear projections on $X$ such that $PQ$ and $QP$ are compact. Does it follow that $PQ$ and $QP$ are finite-rank operators? My attempt: I ...
3
votes
1answer
59 views

Estimate spectral radius of operator product

In my research problem, I have to estimate the spectral radius of the following operator $\chi A$ where $\chi$ is a scalar function taking values 0 or 1 and $A$ is an operator. I can compute ...
1
vote
1answer
62 views

Operator algebra generalization of linear algebra result on diagonalization of commuting operators with distinct eigenvalues

In linear algebra it is true that: a) if $\mathcal{A}$ is a set of unitarily diagonalizable matrices (in $\mathbb{C}$, i.e. normal matrices) that commute with each other then they are simultaneously ...
1
vote
1answer
54 views

Antiderivative as an integral operator from $L^2(0,2\pi)$ to $L^2(0,2\pi)$

I am starting to study Functional Analysis on Hilbert Spaces and I am studying the following operator: $$T:L^2(0,2\pi) \rightarrow L^2(0,2\pi) $$ where $$Tf:(0,2\pi) \rightarrow \mathbb{R} \\ ...
0
votes
0answers
13 views

Let $T$ be a definite integral operator on $(C[a,b])$. Find function $k_j$ such that $T^j (x)=\int_{a}^{t} k_j (s,t) x(s) \,ds$

This is the last part of chain of related questions: I was asked to prove that $$ T:C([a,b]) \to C([a,b]) $$ given by $$ Tx(t)=\int_{a}^{t} x(s) \, ds $$ is linear bounded operator on ...
0
votes
1answer
15 views

linear function, operator norm

Let be $\Phi:V\to W$ a linear function between the vector spaces $V$ and $W$ with the norms $\|\cdot\|_V$ and $\|\cdot\|_W$. Prove that $$\|\Phi\|_{\mathcal{L}(V,W)}=\kappa_{abs},$$ while ...
4
votes
0answers
56 views

Why do these Integration-by-Parts Evaluation Terms Vanish?

The Associated Legendre operator is $$ L_mf = -\frac{d}{dx}\left((1-x^{2})\frac{df}{dx}\right)+\frac{m^{2}}{1-x^{2}}f, $$ where $m$ is a positive integer. For the purposes here, define ...
0
votes
0answers
16 views

Number of solution of $ (\Delta - \lambda) f = \delta $

How they are solutions of the equation $$ (\Delta - \lambda) f (x) = \delta (x)$$ Where $\Delta$ is the Laplacian operator and $\delta(x) = \begin{cases} 0, \quad x\neq 0;\\ +\infty, \quad x= 0 ...
1
vote
2answers
49 views

Can $\text{ arg}$ be thought of as operator?

Forgive me if the question is to vague. The argument, denoted by $\text{arg}$, is a commonly used notation. I am specifically interested in the following use of $\text{arg}$: \begin{align} a=\text{ ...
1
vote
1answer
43 views

Topological characterization of the range of a bounded normal operator

Let $T$ be a bounded normal operator on a Hilbert space $H$. I want to prove the following statement: $\text{ran}(T)$ is closed if and only if 0 is not a limit point of $\sigma(T)$. I tried to use the ...
0
votes
4answers
65 views

Eigenvalues of an operator?

I have just started working with operators, ie objects that map functions to other functions, and I have heard people talking about the eigenavalues of an operator that can be obtained through ...
1
vote
1answer
25 views

Kaplansky density theorem

Let $H$ be a Hilbert space and $A$ a C*-subalgebra of $B(H)$, and $1_H\in A$. Show that the unitaries of $A$ are strongly dense in the unitaries of $\overline{A}^{sot}$. Suppose $U(A)$ be unitaries ...
1
vote
0answers
24 views

Pairs $(p,q)$ such that $id: l_p\to l_q$ is bounded [duplicate]

find all pairs $p,q\in [1,\infty)$ such that $id: l_p\to l_q$ is bounded. This just means I must find all $(p,q)$ such that $\|x\|_q \le C\|x\|_p$ for some $C$ dependent on $p$, $q$. I don't know ...
1
vote
0answers
62 views

Convergence of operator-exponential

Let $T: [0,\infty) \rightarrow L(X)$ define a $C_0$ semigroup on a Banach space $X$, then I want to show that $A_h:=\frac{T(h)-id}{h}$ are such that $e^{tA_h}(f) \rightarrow T(t)(f)$ pointwise. ...
3
votes
1answer
77 views

Frechet Derivatives of a nonlinear integral operator

The nonlinear integral operator $P:C[0,1]\to C[0,1]$ is defined as follow: $$P(f)(x)=1+kxf(x)\int_0^1\frac{f(s)}{x+s}ds$$ In order to obtain the Frechet derivative of the operator, I start with: ...
2
votes
1answer
156 views

Why are eigenfunctions of Laplace-Beltrami operators the minimizer of $\int_\mathcal{M}\| \nabla f(x)\|^2$?

Given a smooth $m$-dimensional manifold $\mathcal{M}$ embedded in $\Re^k$. Suppose we have a map $f : M \to \Re .$ Now, these are my questions: Specific question: i): Why does the $f$ that ...