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
48 views

Self-adjointness

In another thread it was claimed that the operator $O : \operatorname{dom}(O) \subset L^2(-1,1) \rightarrow L^2(-1,1)$ is self-adjoint. $$Of(x)= \frac{f(x)}{{1-x^2}}$$ It is obvious that $$\langle O ...
1
vote
2answers
36 views

Preserve self-adjoint properties

I was thinking about this problem recently: Let $T$ be a self-adjoint operator on $L^2((-1,1),d x)$. Now you define an operator $G$ by $G(f) := T(\frac{f}{(1-x^2)})$ with $\operatorname{dom}(G):=\{f ...
1
vote
1answer
38 views

How to show that the operator $T(\{x_n\})=\{n x_n\}$ has closed graph?

Consider the subspace $$D=\left\{x\in \ell^2 \ \big|\ \sum_{n\in\mathbb N} n^2 |x_n|^2<\infty\right\}$$ of $\ell^2$, and let $T:D\to\ell^2$ be defined by $T(\{x_n\})=\{n x_n\}$. I need ...
0
votes
0answers
16 views

Definition of well-defined for special case

I have a question about what well-defined means in a certain case. For an operator from $X$ to its dual $X^{*}$, say $A:X \rightarrow X^{*}$,why does the definition of $A$ being "well−defined" seem ...
1
vote
1answer
35 views

Vector question involving an operator!

So, here's the problem: An operator H capable of operating on vector x, is defined in terms of a given vector a by: H x=(a * x) where $*$ representes vector product Given that ...
1
vote
0answers
29 views

Definition of nonlinear bounded operator

Hi I am interested in confirming the definition of a bounded operator for a nonlinear operator. Let $X,Y$ be normed spaces. It is well known that a linear operator between $T:X \rightarrow Y$ is ...
1
vote
1answer
27 views

Holomorphic Functional Calculus vs Borel Functional Calculus

I am currently learning about different kinds of functional calculus and I was wondering if I could get something cleared up. The first type of functional calculus we learned about was holomorphic ...
2
votes
0answers
72 views

$C^*$-algebras, von Neumann algebras, unbounded operators and quantum mechanics in connection

I am studying the theory of $C^*$-algebras, von Neumann algebras and unbounded operators in courses on Functional Analysis and Opertor Algebras. Now I want to apply this knowledge to (algebraic) ...
1
vote
0answers
13 views

Strong derivative of a compound map

I find the strong Fréchet derivative of $\Phi(h,\psi(h))$, where $\Phi:T_0\times T_\xi\to Y$ with $T_0, T_\xi, Y$ Banach spaces and $\psi:T_0\to T_\xi$ is strongly differentiable in $0$, evaluated in ...
0
votes
1answer
20 views

Closed restriction of an unbounded self-adjoint operator

Suppose $(A\,;\mathcal{D}(A))$ is an unbounded self-adjoint linear operator (obviously, $\mathcal{D}(A)$ must be dense) on a Hilbert space $\mathcal{H}$. Suppose $\mathcal{D}(C)$ is a proper dense ...
0
votes
1answer
40 views

Iterating average

If $f$ is a continuous function $[0,1]\to \mathbb R$, we define a linear application $T$ as follows $$T(f)(x)=\begin{cases} f(0) & \mathrm{if }~ x=0 \\[0.2cm] \displaystyle ...
2
votes
0answers
35 views

Boundary conditions Legendre equation

I have Legendre's equation $$L(f)=\frac{1}{\sin(\theta)} \left(- \frac{d}{d\theta} \sin(\theta) \frac{df}{d \theta} \right)$$ Now I know that after substituting $\cos(\theta) =x$ we get a ...
1
vote
0answers
29 views

Point spectrum of a nonlinear operator on finite dimensional space

Given a nonlinear operator $T$ mapping $\mathbb R^n$ into itself, are there any known conditions on $T$ ensuring that the number of points in its point spectrum is upper bounded by the dimension $n$?
1
vote
1answer
76 views

Eigenvalues of Left Shift + Right Shift in $l_2([0,\infty))$

This question appeared on an old final exam and I am having difficulty completing it for practice. Let $S_r$ and $S_l$ be defined on the hilbert space $l_2[0,\infty)\to l_2[0,\infty)$ as the ...
1
vote
2answers
107 views

Square root of unbounded operator

Let $T: \operatorname{dom}(T) \subset H \rightarrow H$ be a positive self-adjoint unbounded operator, then I want to define a UNIQUE(!) operator $A$ such that $A^{*}A = T$. Actually, this construction ...
2
votes
0answers
66 views

Continuity of implicit function

I find a proof of the following theorem in A.N. Kolmogorov and S.V. Fomin's Элементы теории функций и функционального анализа (pp. 492-493 here): Let $X,Y,Z$ be Banach spaces, $U$ a neighbourhood ...
0
votes
1answer
36 views

Finite rank approximation of bounded operators on Hilbert space

Let H be a (finite dimensional) Hilbert space. The approximation property states that every bounded operator from H to itself can be approximated by a sequence of finite rank operators. My question ...
1
vote
0answers
37 views

Spectrum of adjoint operator

Let $X$ be a hilbert space and $T\in L(X)$ Show that: (i) $\sigma_c(T^*)=(\sigma(T))^*$ (ii) $\sigma_r(T)=((\sigma_p(T^*))^*)$\ $\sigma_p(T)$ (i): $"\subset"$ Let $\lambda\in\sigma_c(T^*)$ then ...
0
votes
0answers
14 views

operator ranges of a closed linear operator

I have a closed linear operator T on a Hilbert space H. Also I have that range R(T^n) is closed for some n $\in$ N and for non-zero $\lambda$, R(T-$\lambda$I) $\cap$ R(T^n) is closed and ...
1
vote
1answer
42 views

CAR- & Weyl-Algebra: Uniqueness?

Given a Hilbert space: $\mathcal{h}$ Consider representations of the CAR-algebra: $\mathcal{A}_\text{CAR}^{(\prime)}(\mathcal{h})$ In Bratelli & Robinson it is stated the uniqueness: ...
2
votes
1answer
15 views

residual spectra of adjoint operator.

I saw an interesting claim saying "If $λ$ were in the residual spectrum of $T$, then $\overline{λ}$ would be in the point spectrum of $T^*$. Is there a short proof for this claim? recall point ...
0
votes
1answer
37 views

Position Operator on $l^2(\mathbb{Z})$

I'm very familiar with the position operator $(Q\varphi)(x)=x\varphi(x)$ on $L^2(\mathbb{R^d})$, but I'm trying to figure out how to interpret the same operator on $l^2(\mathbb{Z})$ (the space of ...
0
votes
1answer
20 views

Multilateral shift operators

Apparently, the operators $u(e_n) = e_{n+1}$ and $u^\ast (e_n ) = e_{n-1}$ are called "unilateral shift oeprator". Since they have to be called that (instead of calling them "the shift operator") ...
1
vote
1answer
91 views

Spectral Measures: Lebesgue

Preface This thread deals with dominated convergence for functional calculus: $$f_n(\omega)\to f(\omega)\quad(\omega\in\Omega)\implies f_n(E)\to f(E)$$ Framework Given a Borel space $\Omega$ ...
1
vote
1answer
18 views

Approximation in $V^{**}$

We know that for a vector space $V$ there is a natural map $\iota\colon V\to V^{**}$ sending $v$ to $v^{**}$. When $V$ is a normed space, $\iota$ is an isometry, however it may not be surjective. My ...
1
vote
0answers
27 views

tensor products of Banach space

Let $E_{1},\cdots, E_{n}$ be Banach spaces; $n\in\mathbb{N}$ and $\mathbb{R}$ be a real numbers and $E\widehat{\otimes}\mathbb{R}$ be a completion tensor product. We have the fact that ...
1
vote
1answer
30 views

Analytic vectors of self-adjoint unbounded operators

I am working with self-adjoint unbounded linear operators on infinite-dimensional separable Hilbert spaces, and I would like to know some references regarding analytic vectors, especially with respect ...
0
votes
1answer
32 views

Does every continuous operator between normed spaces map bounded sets to bounded sets?

Suppose you have a continuous operator $A$ between two normed spaces $X$ and $Y$. Does it follow that this operator is bounded in the sense that it takes bounded sets to bounded sets, given that $A$ ...
1
vote
1answer
27 views

Measurability of inner integral $x \mapsto \int f(x,y)\, d\mu(y)$

Let $\psi$ be defined by$$\psi(s):=\int_{[a,b]}K(s,t)\varphi(t)d\mu_t$$ where $\varphi\in L_2[a,b]$ and $K\in L_2([a,b]^2)$. Kolmogorov-Fomin's proves the belonging of $\psi$ to $L_2[a,b]$ by showing ...
4
votes
1answer
51 views

Show that $\|B^{-1/2}AB^{-1/2} - I\| >1/2$, when $\|A\|>2\|B\|$

Let $A,B$ be two positive-definite matrices. Suppose that $\|A\|>2\|B\|$. Is it possible to show that $\|B^{-1/2}AB^{-1/2} - I\| >1/2$, where $I$ is an identity matrix and the norm is the ...
4
votes
1answer
46 views

Sturm-Liouville problem and periodic boundary conditions

I was wondering about this: I know that if a 1-d Sturm-Liouville operator is limit circle or limit point then the eigenvalues are simple ( so no degenerated spectrum). But in the case of periodic ...
0
votes
1answer
22 views

Does Unitary must have norm equal to 1?

Does Unitary must have norm equal to 1? I know U is unitary then $UU^*=I$, so $\|Ux\|^2=(Ux,Ux)=(U^*Ux,x)=(x,x)$, so $\|U\|=1$. Is this a proof?
0
votes
1answer
16 views

Existence of adjoint operator in Euclidean space

If we define the adjoint operator of linear operator $A:E\to E$, where $E$ is a complex or real Euclidean, $n$- or $\infty$-dimensional, space, as operator $A^\ast:E\to E$ such that $\forall x,y\in ...
0
votes
1answer
42 views

Two questions about orthogonal projections on Hilbert space

Let $l_{k}^{2}$ denote the k-dimensional Hilbert space and $\oplus_{1}^{\infty} l_{k}^{2}$ be the infinite direct sum of $l_{k}^{2}$. Let $P_{M}\in ...
0
votes
1answer
23 views

Joint spectrum of $\{a_1,…,a_n\}$

Let $\{a_1,...,a_n\}$ be commuting normal operators on a Hilbert space. Put $A:= C^*(1,a_1,...,a_n)$. By Gelfand theorem ,abelian C*-algebra $A$ is identified with the algebra $C(\Omega)$ of all ...
0
votes
1answer
25 views

0 limit point of spectrum of completely continuous operator $H\to H$

I read in Kolmogorov-Fomin's Элементы теории функций и функционального анализа (p. 475 here) that 0 is an accumulation point for the spectrum of a completely continuous operator $A:H\to H$ where $A$ ...
4
votes
1answer
38 views

The set of all normal operators on a Hilbert space is not strongly closed

I need an example to show that the set of all normal operators on a Hilbert space is not strongly closed. Also I know that strong operator topology and strong* operator topology coincide in the set of ...
0
votes
1answer
27 views

Four-indexed infinite matrix

I've found on an ancient book a structure of the kind described below and I do not even know if it is a commonly known structure nowadays. For me it doesn't match neither a direct sum nor a direct ...
2
votes
1answer
79 views

Reiterate Volterra integral operator is a contraction

I read in Kolmogorov and Fomin's Элементы теории функций и функционального анализа (p. 472 here) the statement that Volterra operator $A:L_2[a,b]\to L_2[a,b]$ defined ...
0
votes
1answer
31 views

$\ker (I-A)=\{0\}\Rightarrow\text{im }(I-A)=H$ for $A:H\to H$ compact

Let $T$ be the operator defined by $T:=I-A$ where $I:H\to H$ is the identity and $A:H\to H$ is a compact operator defined on Hilbert space $H$. In such a case, if we defined the chain of ...
1
vote
0answers
33 views

the spectral radius of normal operator

Let $H$ be a Hilbert space and $T$ be linear bounded operator in $H$. Prove that if $T$ is normal then the spectral radius of $T$, $$r(T)=\|T\|.$$ Is this TRUE?
4
votes
1answer
51 views

Green's operator for elliptic differential operator

Let $$ P:\Gamma(E)\rightarrow\Gamma(F) $$ be an elliptic partial differential operator, with index zero and closed image of codimension 1, between spaces $\Gamma(E)$ and $\Gamma(F)$ of smooth sections ...
0
votes
2answers
27 views

$P_M-P_N$ on which subspace of H is orthogonal projection?

Let M and N are two closed subspaces of Hilbert space H such that $N\subset M$. Also $P_M$ and $P_N$ are orthogonal projections on M and N respectively. It is clear that $P_M-P_N$ is again an ...
0
votes
0answers
44 views

Semigroups & Generators: Nonexample: $\delta:\mathcal{C}^1([0,1])\to\mathcal{C}([0,1]):f\mapsto f'$

Given a C*-algebra $\mathcal{A}$. Consider a *-derivation $\delta$. Does it always generate a group: $$\tau(t)=e^{it\delta}$$ But a group of automorphisms is a contraction group: ...
1
vote
2answers
31 views

A proof pertaining to the projector operator

Let $H_{1}$ be any subspace of a Hilbert space $H$, and let $H_{2} = H_{1}^{\bot}$ be the orthogonal complement of $H_{1}$, so that an arbitrary element $h \in H$ has a unique representation of the ...
2
votes
1answer
28 views

Toeplitz Operator is compact if and only if it has finite rank

A referee has pointed out to me that it is "well known that a Toeplitz operator is compact if and only if it has finite rank" and pointed me to R. Douglas: Banach algebra techniques in the ...
0
votes
0answers
19 views

When (weakly) compact operators have pre-adjoints?

Given a bounded linear operator $T\colon X^*\to X^*$ for some Banach space $X$. Then $T$ is an adjoint of an operator $S\colon X\to X$ if and only if $T$ is weakly* to weakly* compact. Are there some ...
2
votes
2answers
91 views

Dense subspaces, closed subspaces and unbounded operators in Hilbert spaces

Let $\mathcal{H}$ be a Hilbert space, and let $N\subseteq\mathcal{H}$. I found two interesting statements (without proof): if a closed subspace $N$ is such that $N^{\perp}=\{0\}$ (which is ...
0
votes
1answer
26 views

How to get the updating rules? after derivation

In the picture i brushed yellow, it dose make no sense to me to get formulas(2) and (3). If anyone could point out or give some references? Thanks a lot!
0
votes
2answers
28 views

Showing that $\mathcal{G}(\ell_2)$ is not dense in $\mathcal{B}(\ell_2)$ via the right shift

This is my question: Is $\mathcal{G}(\ell_2)$ is dense in $\mathcal{B}(\ell_2)$? I am attempting to show that it is not by showing that the right-shift - call it $T:\ell_2 \rightarrow \ell_2$ - ...