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

2
votes
1answer
42 views

A linear map $S:Y^*\to X^*$ is weak$^*$ continuous if and only if $S=T^*$ for som $T\in B(X,Y)$

As the title says, the question is how to prove $S:Y^*\to X^*$ is weak$^*$ continuous if and only if $S=T^*$ for som $T\in B(X,Y)$
3
votes
0answers
53 views

$S$ is continuous with Weak * topology from $Y^*$ to $X^*$ if an only if $S=T^*$ for some $B(X,Y)$ [duplicate]

How to prove that prove that $S$ is weak$^*$-continuous from $Y^*$ to $X^*$ if an only if $S=T^*$ for some $T\in B(X,Y)$ Thanks for any hints. To show that $T$ is continuous is straight forward ...
2
votes
1answer
37 views

What is the relation between the matrix of an operator and that of its adjoint?

Let $X$ and $Y$ be finite-dimensional normed spaces, either both real or both complex, and let $T \colon X \to Y$ be a linear operator. (Then $T$ is bounded since its domain is finite-dimensional). ...
2
votes
0answers
19 views

Prob. 5, Sec. 4.5 in Kreyszig's functional book: The adjoint of the composite of two bounded linear operators

Let $X$, $Y$, and $Z$ be normed spaces, either all real or all complex. Let $T \colon X \to Y$ and $S \colon Y \to Z$ be bounded linear operators. Let $X^\prime$, $Y^\prime$, and $Z^\prime$ denote the ...
2
votes
0answers
62 views

Given $Q:ℝ^d→(\text{Hilbert-Schmidt operators }U→ℝ^d)$, find a Hilbert-Schmidt operator $T:U→L^2(ℝ^d,ℝ^d)$ with $Q(x)u=(Tu)(x)$

Let$^1$ $U$ be a separable $\mathbb R$-Hilbert space $d\in\mathbb N$ $\lambda$ denote the Lebesgue measure on $\mathbb R^d$ $\Omega\subseteq\mathbb R^d$ be a bounded domain $H:=L^2(\Omega,\mathbb R^...
1
vote
0answers
33 views

How to prove $n(T)=\sup\{|\langle Tx,x \rangle |, \|x\|=1\}$ is a norm on $B(H)$ and $n(T)\lt\|T\|\lt2n(T)$ where $T\in B(H)$? [closed]

Let $H$ be a Hilbert space over $\mathbb C$. If $T\in B(H)$, how to prove that $$n(T)=\sup\{|\langle Tx,x \rangle |, \|x\|=1\}$$ is a norm on $B(H)$ and $$n(T)\lt||T||\lt2n(T)\ \textrm{?}$$ I couldn'...
1
vote
1answer
20 views

Applications of Positive Operator Valued Measures (POVMs)

I am wondering what some applications of POVMs are in mathematics (or mathematical physics)? I am going through Berberian's 'Notes on Spectral Theory', which shows how we can write a normal operator ...
1
vote
2answers
90 views

Restriction of operators on $l_\infty$ to $c_0$

Given $\epsilon>0$, can we always find a non-compact operator $T:l_\infty\to l_\infty$ of norm larger than $1$ such that the restriction of $T$ to $c_0$ is compact and has norm smaller than $\...
1
vote
1answer
23 views

Continuous inverse of an unbounded operator on a Hilbert space

Let $T:D(T)\to H$ be an unbounded densely defined operator on a Hilbert space $H$. Suppose that $T^{-1}$ is continuous, i.e. that $0$ belongs to the resolvent set $\rho(T)$ of $T$. As $T^{-1}$ exists,...
1
vote
1answer
47 views

Norms are equivalent iff dual spaces for them are the same?

It is trivial that if we have a vector space $X$ and two equivalent norms on it than $X'_1$ -dual space (of continuous functionals) for the first norm and $X'_2$ are the same spaces. Is the converse ...
0
votes
0answers
19 views

Proving that $-\Delta+V$ on some domain is self-adjoint

This question may look as a "proof-reading" question, but what I ask is if I correctly understand the way these concepts work, by showing how I think about them. Suppose I have the following three ...
1
vote
1answer
37 views

operator inequality using spectral theorem

Given two densely defined unbounded self-adjoint strictly positive operator $A$ and $L$ in Hilbert space $H$ with domain $D(A) \subset D(L)$ and $\|Lx\| \leq \|Ax\|$ for all $x\in D(A)$, why do we ...
2
votes
0answers
30 views

Boundedness of a naive integral operator

Define integral operator $J:L^2[0,1] \to L^2[0,1]$, $$Jf(x) := \int_0^x f(s) ds.$$ I am wondering if the following equivalence holds, $\|Jf\|_{L^2} \simeq \|f\|_{H^{-1}}$, where $\|\cdot\|_{H^{-1}}$ ...
0
votes
1answer
10 views

A restriction $A_2$ of a compact self-adjoint compact linear operator $A$ is also compact and self-adjoint?

Let $X$ be an inner product space and let $A$ be a compact and self-adjoint linear operator. Let $p_1$ be an eigenvector of $A$. Let $A_2$ be the restriction of $A$ to $X_2$ where $X_2$ is given by $$...
1
vote
1answer
27 views

Question on spectral theorem for compact operators

I'm studying a proof of the spectral theorem for compact operators. The first part of it reads as follows: Let $X$ be an infinite dimensional inner product space and let $A: X \to X$ be a compact and ...
0
votes
1answer
24 views

Operator commutes with spectral projection

Let $E$ be the spectral measure to an (unbounded) self-adjoint operator $A$. Is there a sufficient and necessary condition so that for a bounded interval $I$ we have $E_I A= AE_I$?
0
votes
1answer
27 views

Proof of matrixnorm relative to one-norm

Let $A \in \mathbb{R}^{d \times d}$ a $d \times d$-matrix $A=(a_{ij})$ with norm $\|\cdot\|_1$. Proof: $$\|A\|= \max_\limits{j=1,...,d} \sum_\limits{i=1}^d |a_{ij}|$$ Let $\|x\|_1=1$ and $Ax=y$: $\|...
0
votes
1answer
55 views

what is the reason of this of the following statement?

in a paper i saw the following statement: Let $\Phi:B(X)\longrightarrow B(X)$ is an additive and surjective map. If $T\in B(X)$ and for some $x\in X$ $Tx \otimes {\Phi(T)}^*f=\Phi(T)x\otimes T^*f$ ...
1
vote
0answers
27 views

Prove for the family of operators $\{S(t)\}_{t>0}$ that $S'(0)$ exists

Let's consider a real-valued function $V\in L^\infty(\mathbb{R})$ and a family of operators $\{S(t)\}_{t>0}$ defined on $L^2(\mathbb{R})$ as follows: $$\qquad \qquad\left(S(t)f \right)(x) = \frac{...
2
votes
1answer
40 views

Is this a general identity for the Resolvent? [solved: integral representation of the resolvent]

And if so, what is it called? $$i(H-\lambda - i\epsilon)^{-1}\phi = \int_0^\infty e^{-\epsilon t} e^{i\lambda t}e^{-iHt}\phi\,\text dt$$ as in Reed-Simon XIII.7 example 1. It is stated there for $H=-i\...
1
vote
2answers
74 views

Integrating a Linear Operator $A:H\longrightarrow H$ (Matrix)

I am trying to prove a functional analysis proposition, but I got stuck. I have to integrate a matrix. In my proof I use the following matrix: Let $A$ be a self-adjoint matrix on $H=\mathbb{C}^n$ ...
4
votes
1answer
29 views

Non-self Adjoint Operator Algebra References

The problem I am working on has led me to define a norm closed sub-algebra $\mathscr{A}$ of $\mathscr{B}(\mathscr{H})$. The algebra is generated by some mild relations, and in general, will not be ...
0
votes
0answers
41 views

Injective Integral Operator on $L^2[0,1]$ or $C[0,1]$?

Consider an arbitrary $f \in L^2 [0,1]^+ $ where $L^2[0,1]^+$ is the function space of square integrable non negative functions. We say $T$ is an Integral Operator if $T$ is of the form , $$ T(f) = ...
1
vote
0answers
72 views

Calculate the trace of $LBB^*$, where $L:H→H$ and $B:=ΦT^{1/2}$ for some $Φ:U_0→H$, an embedding $ι:U_0→V$ and $T:=ιι^*$

Let$^1$ $U$, $V$ and $H$ be $\mathbb R$-Hilbert spaces $Q\in\mathfrak L(U)$ be nonnegative and symmetric $U_0:=Q^{1/2}U$ be equipped with $$\langle u,v\rangle_{U_0}:=\langle Q^{-1/2}u,Q^{-1/2}v\...
0
votes
0answers
13 views

Application of Uniform boundedness theorem: $\langle Tx,y\rangle$ bounded for each $x,y$ then $||T||$ is bounded

For Hilbert Space $X$, if we have a condition on a subset $F\subset B(X)$ ('set of bounded linear operators on $X$') such that $$ \{\langle Tx,y\rangle:T\in F\} $$ is a bounded set for each $x,y\in ...
0
votes
0answers
25 views

Eigenvalues of Finite Type

I want to show that the following holds: Let $T:X\rightarrow Y$ and $S:Y\rightarrow X$ be operators acting between Banach spaces. Assume that $\mu \not=0$ is an eigenvalue of finte type of $ST$. ...
1
vote
1answer
24 views

Characterization of orthogonal projections in terms of operator norms

I want to show the following equivalence: If $X$ is a Hilbert Space and $P\in B(X)$ (i.e. $P$ is bounded and linear) and $P^2=P$, then $$ (\text{im}\,P)^{\perp} =\ker P\iff ||P||\le 1 $$ I know that ...
4
votes
1answer
37 views

Spectrum of difference of two projections

Let $p$ and $q$ be two projections in a $C^*$-algebra. What can one say about the spectrum of $p-q$, i.e. is $\sigma(p-q) \subset [-1,1]$ ? The exercise is to show that $\lVert p-q \rVert \leq 1$. ...
0
votes
2answers
31 views

$A$ is a Hermitian operator on an infinite dimensional Hilbert space and $\langle Ax|x\rangle=0$ for all $x$, prove $A=0$ without the spectral theorem

If $A$ is a Hermitian operator on an infinite dimensional Hilbert space such that $\langle Ax|x\rangle=0$ for all vectors $x$, can we prove $A=0$ without the spectral theorem? The proof seems ...
1
vote
0answers
10 views

Spectral bands of periodic differential operators

I am reading the book "Multidimensional Periodic Schrödinger Operator" (O. Veliev, 2015) which says on page 11: It is well-known the following statements about the spectral properties of $L_{t}(q)$...
0
votes
1answer
28 views

If $A_j$ is an increasing family of Hermitian operators such that $A_j\nearrow A$ weakly, for $A=\mathrm{LUB}A_j$, then $A_j\rightarrow A$ strongly.

I am trying to prove the following proposition from Berberian's 'Notes on Spectral Theory': Proposition 1: If ($A_j$) is an increasingly directed family of Hermitian operators, and if the family ...
1
vote
1answer
51 views

Elementary proof that $a - 1$ is not invertible, for self-adjoint $a$ with $\lVert a \rVert = 1$

Assume $a \in A$ where $A$ is a unital $C^*$-algebra. If $\lVert a \rVert = 1$ and $a^*=a$ we know that $1 \in \sigma(a)$, the spectrum of $a$. This follows from the fact that $\lVert a \rVert = r(a) =...
2
votes
1answer
53 views

How can we compute the square root of an operator of the form $Cv=\sum_{n\in\mathbb N}\langle v,e_n\rangle_Ve_n$?

Let $\mathbb K\in\left\{\mathbb C,\mathbb R\right\}$ $U$ and $V$ be $\mathbb K$-Hilbert spaces such that $U\subseteq V$ and that the inclusion $\iota$ is Hilbert-Schmidt $C:=\iota^\ast$ denote the ...
0
votes
1answer
58 views

How can we compute the adjoint of the inclusion between two Hilbert spaces?

Let $\mathbb K\in\left\{\mathbb C,\mathbb R\right\}$ $(U,\langle\;\cdot\;,\;\cdot\;\rangle_U)$ and $(V,\langle\;\cdot\;,\;\cdot\;\rangle_V)$ be $\mathbb K$-Hilbert spaces such that $U\subseteq V$ ...
2
votes
0answers
51 views

I have an embedding $\iota$ between two Hilbert spaces and want to know if $\iota\iota^\ast$ is something simple like an orthogonal projection

I'm reading A Concise Course on Stochastic Partial Differential Equations. In Proposition 2.5.2 the authors define the notion of a cylindrical $Q$-Wiener process $W$. It turns out that $W$ is just a $...
0
votes
1answer
29 views

Inversible operator in Hilbert space

Consider $\phi\in L^{\infty}[0, 2\pi]$. Let M be operator $L_2[0, 2\pi]\rightarrow L_2[0, 2\pi]$$$Mf = \phi f$$ In $L_2[0, 2\pi]$ we have topological basis ${e^{inx}}, n\in \mathbb Z$. $L_2[0, 2\pi] =...
2
votes
1answer
36 views

spectrum of an operator restricted to an invariant subspace

Let $X$ be an infinite-dimensional real Banach space and $T\in\mathcal{L}(X)$ a continuous linear operator acting on $X$. Suppose $W$ is a finite-codimensional $T$-invariant closed subspace of $X$, ...
1
vote
1answer
36 views

Let $H$ be a Hilbert space, $V≤H$ be closed, $Q:H→V$ be the orthogonal projection, $(e_n)_{n∈ℕ}$ be an ONB of $H$. Is $(Qe_n)_{n∈ℕ}$ an ONB of $V$?

Let $\mathbb K\in\left\{\mathbb C,\mathbb R\right\}$ $U$ and $H$ be $\mathbb K$-Hilbert spaces $(e_n)_{n\in\mathbb N}$ be an orthonormal basis of $H$ $\iota:U\to H$ be an embedding and $V:=\iota(U)$ ...
2
votes
1answer
66 views

Find the spectrum of an operator

I am trying to learn some basic stuff about spectral theory, and I am a little bit lost. Please, could you help me and tell me how to find $\sigma(T)$ and $\sigma_p(T)$ of the operator $T:C([0,1]) \...
0
votes
1answer
31 views

Linear Operator on Hilbert Space $l(\mathbb Z)$

Let $A$ be the linear operator on $l(\mathbb Z)$ defined for $u=\{u_k\}_{k \in \mathbb Z}$ as $(Au)_k = \sum_{h=-\infty}^{+\infty}a_{k,h}u_h$ where $a_{k,h}=\frac{1}{(k-h)2}$ for $h \not= k$, and $...
1
vote
0answers
49 views

Linear Operators on $L_2(\mathbb R)$ definfed as Integrals

Let's consider the linear operators on $L_2(\mathbb R)$ $$ T_{\alpha}f(x) = \int_{-\infty}^{+\infty} \frac{e^{-|x-y|^2}}{(1+x^2)^{\alpha}}f(y)dy $$ with ${\alpha} \in [0,1]$. Find ${\alpha}$ such ...
3
votes
0answers
37 views

Square-root of $\iota\iota^\ast$, where $\iota$ is an isometric embedding between Hilbert spaces

Let $U$ and $H$ be Hilbert spaces and $\iota$ be an embedding of $U$ into $H$. Then, $$\pi x:=u\;\;\;\text{for }x\in H\text{ with }x=\iota u+y\text{ for some }u\in U\text{ and }y\in\left(\iota U\right)...
1
vote
1answer
35 views

Dimension of operators

Question: Let $T: \ell^2 \to \ell^2$ be self-adjoint and compact. For $\lambda \in \mathbb{R}$, let $$ S_\lambda = \overline{\text{Span} \{ v \in \ell^2 \mid Tv = \gamma v \text{ for some } \gamma \le ...
2
votes
1answer
24 views

The point spectrum and residual spectrum of an operator on $l_2$ related to backward shift

I have a problem with the spectrum of this operator: $(Tx)_1 = x_2$ $(Tx)_2 = x_1$ $(Tx)_n = \frac{1}{n}x_{n+1}$ with $n\ge3$ Find the $||T||$, the point spectrum $\sigma_P(T)$ and $\sigma_P(T^{\...
0
votes
0answers
25 views

Nullspace of strange operator

I have the following equation: $0=\frac{\partial}{\partial y}(e^{-\beta U(x,y)}\frac{\partial}{\partial y}(P(x,y,t)e^{\beta U(x,y)}))$ and would like to study its solvability (Fredholm) conditions (...
0
votes
0answers
30 views

Given a special Hilbert space $U_0$, is there a proper superspace $V$ such that the inclusion $\iota:U_0\to V$ is Hilbert-Schmidt?

Let $U$ be a Hilbert space $Q\in\mathfrak L(U)$ be nonnegative and symmetric $U_0:=Q^{1/2}U$ be equipped with $$\langle u,v\rangle_{U_0}:=\langle Q^{-1/2}u,Q^{-1/2}v\rangle_U\;\;\;\text{for }u,v\in ...
1
vote
1answer
33 views

Can we find a concrete representation of $\iota\iota^\ast y$, if $\iota$ is a Hilbert-Schmidt embedding between Hilbert spaces?

Let $U$ and $H$ be real Hilbert spaces $\iota:U\to H$ be a Hilbert-Schmidt embedding $Q:=\iota\iota^\ast$ Can we find a concrete representation of $Qy$ for some $y\in H$? By Riesz' ...
0
votes
0answers
27 views

Complemented Spaces: Continuity vs. Closedness

Given a topological vector space $X$. (Not necessarily Hausdorff!) Consider subspaces: $$U_\pm\leq X:\quad X=U_++U_-\quad U_+\cap U_-=(0)$$ Equivalently an isomorphism: $$\Phi:U_+\oplus U_-\...
1
vote
1answer
28 views

Closable Operators: Nonexample

Given the Banach space $X:=\mathcal{C}([0,1]\cup[2,3])$. I remember I've seen a beautiful example of a non-closable operator whose graph is dense. It involved exploiting Stone-Weierstraß for a ...
0
votes
1answer
22 views

Hahn-Banach: Operators

Given two Banach spaces $X$ and $Y$. (More generally locally convex spaces) Regard a closed subspace $U\subseteq X$. Does every bounded operator extend: $$T\in\mathcal{B}(U,Y)\implies T_E\in\mathcal{...