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.
9
votes
0answers
95 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 ...
8
votes
0answers
192 views
Limit of sequence of growing matrices
Let
$$
H=\left(\begin{array}{cccc}
0 & 1/2 & 0 & 1/2 \\
1/2 & 0 & 1/2 & 0 \\
1/2 & 0 & 0 & 1/2\\
0 & 1/2 & 1/2 & 0
\end{array}\right),
$$
...
8
votes
0answers
177 views
Inverse of Toeplitz Matrix Property
Sorry if this question has been asked already but I didn't find it. Given a symmetric Toeplitz matrix of the form
$$\left[\begin{array}{llll}
a_0 & a_1 & \dots & a_n\\
a_1 & a_0 ...
6
votes
0answers
149 views
Hilbert transform and Hilbert matrix
The Hilbert matrix is
\begin{bmatrix}
1 & \frac{1}{2} & \frac{1}{3} & \frac{1}{4} & \dots \\[4pt]
\frac{1}{2} & \frac{1}{3} & \frac{1}{4} & & \ddots \\[4pt]
...
6
votes
0answers
192 views
Min Max Principle and Rayleigh-Ritz-Method for eigenvalues of unbounded operators?
Finding eigenvalues of matrices using the Rayleigh-Ritz quotient is well-known, c.f. http://en.wikipedia.org/wiki/Min-max_theorem
Does the following generalization of that fact also hold?
Theorem: ...
6
votes
0answers
109 views
Invertibility of Toeplitz operator in $\ell_1$
Suppose we have a Toeplitz operator
$$
T(a) = \begin{bmatrix} a_{0} & a_{-1} & a_{-2} & \ldots & \ldots &a_{-n+1} &\dots \\\\
a_{1} & a_0 & a_{-1} & \ddots & ...
5
votes
0answers
49 views
When are two commuting linear operators functions of each other
I've computed that the following is valid for certain functions but I've hit a slight bump in my proof. I was wondering if someone could clear it up.
If we formally consider the integral operators ...
5
votes
0answers
133 views
Question about the Spectral Theorem for Self Adjoint Operators and Eigenvalues
I have been working through Teschl's book "Mathematical Methods in Quantum Mechanics with Applications to Schrodinger Operators" and I am stuck on a problem in Chapter 3. I am trying to prove that if ...
5
votes
0answers
189 views
Why is the numerical range of an operator convex?
Let $T$ be a Hilbert space operator. Its numerical range is \begin{equation} W(T)=\{\langle Tx,x\rangle:\|x\|=1\}.\end{equation}
It is a well-known fact that $W(T)$ is a convex subset of the complex ...
5
votes
0answers
117 views
Two “different” adjoints of exterior derivative on manifolds with boundary in the $L^2$-setting
The follow problem appears in the setting of $L^2$-differential forms on manifolds with boundary. An abstracted operator theoretic problem is given below.
Suppose $M$ is a smooth Riemannian manifold ...
5
votes
0answers
184 views
Where does the notation $\mathrm{Ad}(U)$ for $a\mapsto UaU^*$ come from?
I have often seen, in the context of operator theory and operator algebras, the notation $\mathrm{Ad}(U)a=UaU^*$, where $U$ is a unitary operator on a Hilbert space $H$ and $a$ is a bounded linear ...
4
votes
0answers
44 views
Concerning unbounded linear operators on a Hilbert space
Let $H$ be some Hilbert space and let $B:H\rightarrow H$ be a bounded linear operator and $T:H\rightarrow H$ an unbounded linear operator. Furthermore we assume that $T$ is closed ,i.e. it's graph in ...
4
votes
0answers
119 views
Inverse of Identity plus Volterra operator
consider the following operator or $L_2(0,1)$,
$(Pw)(x)=w(x)+\int_0^x K(x,y)w(y)dy+\int_x^1 K(y,x)w(y)dy$, where the integral kernel is a polynomial.
I am trying to construct the inverse of this ...
4
votes
0answers
120 views
Prove that the integral operator is bounded
Prove that the following operator is bounded on $L^{2}(0, \infty)$:
$Af(x)$ = $\frac{1}{\pi} \int_{0}^{\infty} \frac{f(y)}{x+y}dy$
with $||A|| \le 1$.
Attempt at Solution
It can be shown that:
...
4
votes
0answers
85 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 ...
4
votes
0answers
71 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 ...
4
votes
0answers
218 views
Confused by a proof in Rudin *Functional Analysis*
I am reading Rudin's Functional Analysis and got quite confused by his proof of Thm 8.5, that is, the existence of fundamental solutions for differential operator $P(D)$, where $P$ is a polynomial.
...
4
votes
0answers
197 views
Spectral theorem for unitary operators
I saw in several texts, as a part of the spectral theorem for unitary operators, that given a unitary operator $U$ on a Hilbert space $H$ (say it is separable), $H$ can be decomposed as an orthogonal ...
4
votes
0answers
185 views
Sum of operator and adjoint is self-adjoint
In abstract Hodge theory there is the following lemma:
Let $H$ be a Hilbert space and $A \in \mathcal{C}(H)$ a densely defined, closed operator (so possibly unbounded) and $A^*$ its adjoint operator. ...
4
votes
0answers
109 views
Does such an operator exist?
Suppose $T$ is a bounded operator on $H:=\mathcal{l}_2$ which is quasinilpotent and has the property that both $T$ and $T^{*}$ are not injective and have finite dimensional kernels.
Is it possible ...
4
votes
0answers
182 views
Eigenprojection as Contour Integral over Resolvent
Let $H$ be a Hilbert space and let $A \in L(H)$ be a bounded linear operator. Assume that $\lambda$ is an eigenvalue of $A$ and assume further that $C_\lambda$ is a simple closed curve in the complex ...
4
votes
0answers
63 views
Relations between spectrum and quadratic forms in the unbounded case
Let $H$ be a complex Hilbert space. If $B$ is a bounded self-adjoint operator on $H$ then its spectrum is a closed and bounded subset of the real line and we can find its extremes in terms of the ...
4
votes
0answers
63 views
Extensions of finite-rank operators
Let $V$ be a vector space and let $W$ be its subspace of infinite codimension. Let $\mathcal{F}_W$ be the family of all finite-rank operators on $V$ with range contained in $W$. Consider the ...
4
votes
0answers
67 views
change of spectrum under diagonalisation
I have the following question. Let $T\colon \mathcal{H} \to \mathcal{H}$ be a bounded, self-adjoint operator on a Hilbert-space $\mathcal{H}$. By spectral theorem we know that there exists a measure ...
4
votes
0answers
191 views
Fixed point: linear operators
I ask my question in two parts: though the topic is similar, I would like to distinguish linear and general cases since methods may be too different while my questions are broad.
Consider a space $X$ ...
4
votes
0answers
296 views
Double dual of the space $C[0,1]$
The second dual or double dual of the space of all continuous functions on $[0,1]$, $C[0,1]$ is von Neumann algebra. Can anyone help me identifying this space?
4
votes
0answers
137 views
Set of all compact operators $K(H)$ is the unique ideal in $B(H)$?
I want to show that the set of all compact operators $K(H)$ is the unique ideal in $B(H)$. Is there any relation between invertibility and compactness of an operator?
3
votes
0answers
65 views
Conditions for a kernel of a bounded operator to be complemented
I am well aware of the problem of complementing subspaces in Banach spaces as it was discussed here and here .
Nevertheless, I wonder whether there are conditions for existence of a complement $M$ ...
3
votes
0answers
46 views
Sub-unital maps between C*-algebras: is there any relevant result?
"In this section, we deal with positive linear maps $\phi : A \rightarrow M$ between two unital C∗-algebra $A$ and $M$ with units denoted by $I$. In fact, we may assume that $A$ is the unital ...
3
votes
0answers
70 views
Is the sum of two closed operators closed?
If A and B are closed linear operators from $X$ to $X$ ($X$ is a normed vector space and the domain of them is X), is $A+B$ a closed operator? I think it's not but I can't find a counterexample.
In ...
3
votes
0answers
75 views
The norm of an operator
Let $\rho(x)$ be a weight function in a unit sphere, such that
\begin{equation}
\begin{array}{l}
\displaystyle 1. \rho(x)\ge 0,\int_{\mathbb{R}^n}\rho(x)=1\\
\displaystyle 2. \rho(x)\in ...
3
votes
0answers
111 views
show that the function satisfies condition of the lemma
Let $(f_n)_{n\geq 0}$ be an eigenfunctions of the compact integral operator
$F$, defined on $L^2([-1,1])$ by
$$
F(f)(x)=\int_{-1}^1\frac{\sin a(x-y)}{x-y}\,\psi(y)\,\mathrm{d}y, \quad \mbox{here ...
3
votes
0answers
68 views
invertible operator Sobolev space
Let $T:H_0^2(D)\rightarrow H_0^2(D)$ a linear bounded operator defined via the sesqulinear form $(Tu,v)=a(u,v):=\int_{D}\Delta u\Delta v dx$. I have shown that T is coercive and self-adjoint. How can ...
3
votes
0answers
56 views
Application of a result on some bounded functionals on a subspace of $C([0,1])$
The following result was proved in a previous post:
Bounded functionals on Banach spaces.
Let $(X, \|.\|)$ be a Banach space such that
$X \subset C([0,1]) $
For every $r\in \mathbb{Q}\cap[0,1], ...
3
votes
0answers
106 views
Two questions about ultraweak and ultrastrong topology from Dixmier
You could reference Dixmier's book on Von Neumann Algebras p.42 Theorem 1 and its proof to know the entirety of the context. Otherwise, the most relevant things are below:
Let $M$ be an ultraweakly ...
3
votes
0answers
147 views
surjectivity of operators on $\ell^\infty$
Can anyone give me an example of an operator $T:\ell^\infty\rightarrow\ell^\infty$ which has dense range but is not surjective?
3
votes
0answers
122 views
Find spectrum of $Af (x) = \int\limits_0^\pi \sum\limits_{n=1}^\infty 5^{-n} \cos(nx)\cos(nt) f(t)\;dt$
I need to find spectrum of $Af (x) = \int\limits_0^\pi \sum\limits_{n=1}^\infty 5^{-n} \cos(nx)\cos(nt) f(t)\;dt$ in $L_2[0,\pi]$.
I know that this operator is self-adjoint, so its residual spectrum ...
3
votes
0answers
111 views
Construct a multiplication operator which has dense point spectrum
By a multiplication operator here we mean an operator
$$Af(t)=m(t)f(t), \qquad f \in D(A)=\{x \in L^2(\mathbb{R} \mid m(t)f(t) \in L^2(\mathbb{R})\}$$
where $m$ is a Borel measurable function on ...
3
votes
0answers
395 views
Continuous spectrum can shrink to an isolated point
Let $A$ be a bounded linear operator in a Hilbert space $H$.
I had the misconception that the continuous spectrum of $A$ would necessarily have some "continuous" appearance: an interval, a union of ...
3
votes
0answers
108 views
Fixed point: general case
This is the second part of the question Fixed point: linear operators. Here I would like to ask you about the general case.
A lot of concepts can be described or even defined as a solution of a ...
3
votes
0answers
126 views
Cramer's rule for infinite dimensional vectors
For the equation $Ax = b$ in the finite dimensional linear space one can apply Cramer's rule to find $x$ if operator $A$ is linear. If there is an equivalent or a similar method for an infinite ...
2
votes
0answers
57 views
Why matrix representation of convolution cannot explain the convolution theorem?
A record saying that Convolution Theorem is trivial since it is identical to the statement that convolution, as Toeplitz operator, has fourier eigenbasis and, therefore, is diagonal in it, has ...
2
votes
0answers
61 views
Relation between noncommutative geometry and functional analysis
Recently I came across the subject of noncommutative geometry via my interest in functional analysis. My very little exposure to this subject gives me a sense that part of it is built on the theory of ...
2
votes
0answers
113 views
How to decompose a representation into direct sum of cyclic representation?
Let $U$ be the bilateral shift operator on $l^2 (\mathbb Z)$, let $T=U+U^*$. How to calculate $\sigma(T)$? And how to show there is no cyclic vector for the action of $C^*(T,I)$. Further how to ...
2
votes
0answers
39 views
Doubt about the spectrum of an operator
I consider the Laplacian operator
$$A=-\Delta$$ in the domain $$H^2(\mathbb{R}^3)$$ where it is selfadjoint. We know that its spectrum is $[0,+\infty)$. Now I want to consider the restriction of $A$ ...
2
votes
0answers
70 views
Find the adjoint operator
I would like to find the adjoint operator in the Hilbertspace $L^2(0,\infty)$ of the operator
$$
(Ax)(t)=x(at), x\in L^2(0,\infty), a>0.
$$
My calculation is the following; I use the ...
2
votes
0answers
129 views
Fredholm and Compact Operators
Let $X$ and $Y$ be Banach spaces and $T\in B(X,Y)$ be Fredholm. Then there is $S\in B(Y,X)$ such that $ST=I+K_{1}$ and $TS=I+K_{2}$ where $K_{1},K_{2}$ are compact operators.Proof: Since $T$ is ...
2
votes
0answers
48 views
Show compactness of an evolution operator
Consider the heat equation
$$
u_{t}=u_{xx},~~~~~u_0(x)=u(0,x)$$
with $u\colon [0,T]\times\mathbb{R}\to\mathbb{R}, (t,x)\mapsto u(t,x)$
and the evolution operator $E(T)$ with $E(T)u_0=u(T,x)$.
1.) ...
2
votes
0answers
125 views
Determining the spectral representation of a operator
The spectral representation for a self-adjoint operator $T \in L(H)$ for H a Hilbert space is written as:
$$
T = \sum_{\lambda \in \sigma(T)} \lambda \pi_{\lambda},
$$
where $\sigma(T)$ is the ...
2
votes
0answers
56 views
Verify a given SVD of an operator
Show that the Singular Value Decomposition of the operator
$$
A\colon L^2([0,1])\to L^2([0,1]), x\mapsto\int\limits_0^t x(s)\, ds
$$
is given by
$$
...
