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

0
votes
0answers
4 views

Unclear passage of a theorem concerning compact operators (Schauder fixed point theorem)

I'm looking at this proof of Schauder theorem and I am struggling with a passage. This is my problem: Let $X$ be a Banach space, $K \subset X$ a convex, close and bounded set and $F:K \rightarrow ...
0
votes
0answers
12 views

Does a pseudodifferential operator $A$ commute with the Resolvent of $A^*A$?

Suppose we have a pseudodifferential operator $A$ (not necessarily bounded) for which the heat operator $$ e^{-tA^*A} := \frac{i}{2 \pi} \int_\Gamma e^{-t\lambda} (A^*A - \lambda)^{-1} \;d\lambda ...
0
votes
0answers
13 views

Finite dimensional C*-algebras and spectrum of each its elements

Let $A$ be a finite dimensional C*-algebra. when I say finite dimensional I mean there is $x_1,...,x_n\in A$ such that $A=span\{x_1,...,x_n\}$(C*- algebra generated by $\{x_1,...,x_n\}$)(is it ...
0
votes
1answer
17 views

GNS Construction on non-unital algebra

STATEMENT: If A has a multiplicative identity 1, then it is immediate that the equivalence class $ξ$ in the GNS Hilbert space H containing 1 is a cyclic vector for the above representation. If $A$ is ...
1
vote
1answer
16 views

Showing an Operator is Well-Defined and Bounded

Let $\{e_n\}_{n \in \mathbb{N}}$ be an orthonormal system within $\ell^2$. Fix a sequence $\lambda = (\lambda_1, \ldots , \lambda_n , \ldots) \in \ell^{\infty}$ and define $ \displaystyle Tf = ...
2
votes
0answers
26 views

Computing the Spectrum of an integral operator

Let $K:L^2([0,1])\rightarrow L^2([0,1]),$ $(Kx)(t)=\int_0^tx(s)ds$. Now I have to compute the spectrum. But I don't have any idea how to do this. If I would know when $(K-\lambda)$ is bijective then ...
0
votes
0answers
15 views

Finding the adjoint operator

On the interval $(0,1)$ consider the differential operator $Lu=u''''+u'$ with boundary conditions $u(0)+u'(1)=u(1)+u'(0)=0$ $2u(0)+u''(1)=2u(1)+u''(0)=0$ $(1)$ I want to find the adjoint ...
0
votes
0answers
11 views

Modulus: Invariant Domain

Problem Given Hilbert spaces. Consider a bounded operator: $$M:\tilde{\mathcal{H}}\to\mathcal{H}:\quad \|M\|=1$$ Regard its minimal polar decomposition: $$\tilde{\mu}:=\sqrt{M^*M}:\quad ...
2
votes
1answer
14 views

Compactness of translation operator in weighted spaces

Let $x,v\in\Bbb R^d$, $t\in \Bbb R$ and $m(x,v)$ be a smooth strictly positive function rapidly decaying on infinity - think $m(x,v) = \exp(-|x|^2-|v|^2)$. Define Banach spaces $X$ and $Y$ by ...
0
votes
1answer
25 views

Finite dimensional C*-algebra

Let $A$ be a simple and finite dimensional C*-algebra. We first note that $aAb\neq 0$ for every nonzero $a,b\in A$. Let $B$ be a maximal abelian self-adjoint subalgebra of $A$. Being finite ...
0
votes
0answers
22 views

Positive element of a C*-algebra

Let $A$ be an abelian C*-algebra and $p$ be a projection in $A$. To show $p$ is an extreme point of $A^+_{\|.\|\leq 1}$ suppose there is $b,c\in (A^+)_{\|.\|\leq 1}$ such that $p= \frac{1}{2}(b+c)$ ...
0
votes
1answer
29 views

Domain of operator via spectral theorem

Assumptions: Suppose $T$ is a self-adjoint operator (possibly unbounded) from the (dense) domain $D(T)$ on a Hilbert space $H$, hence $T:D(T)\rightarrow H$. Assume that $f$ is a continuous function on ...
1
vote
0answers
17 views

Spectral Measures: Helffer-Sjöstrand

Given a Hilbert space. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ Then one has the Helffer-Sjöstrand formula: $$f\in\mathcal{C}^\infty_0(\mathbb{R}):\quad ...
1
vote
1answer
31 views

Spectral Measures: Stone's Formula

Given a Hilbert space. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ Then Stone's theorem says: ...
2
votes
2answers
27 views

Product unbounded operators

Let $A : D(A) \subset H \rightarrow H$ be unbounded and $B$ be a bounded operator, both of them are self-adjoint, then $(AB)^* = B^*A^*$ and $(BA)^* = A^*B^*$, right? I just wanted to be sure that ...
0
votes
1answer
36 views

Why if $T$ is not a bounded operator then exists $ (x_n) $ that converges to $ 0_{X} $ for which $ \| T(x_n) \| \geq n^2 $ for all $ n $?

Let $X$ and $Y$ be normed spaces. Suppose that $ T: X \to Y $ is a linear operator and assume that $T$ is not bounded. Why with these assumptions can I say that exists a sequence $ (x_{n})_{n \in ...
0
votes
1answer
46 views

Spectral theorem for unbounded self-adjoint operators, questions about the proof

I want to understand the proof of the Spectral theorem for unbounded self-adjoint operators. First the theorem: Let H be a separable complex Hilbert space, $A:D(A)\subseteq H\to H$ a densily defined ...
0
votes
1answer
20 views

If $A$ is an abelian C*-algebra, and $\tau$ is pure then it is a character on $A$

If $A$ is an abelian C*-algebra,and positive linear functional $\tau$ is pure then it is a character on $A$. Murphy in his book(C*-algebras and operator theory) has below proof: While I think we can ...
1
vote
1answer
26 views

Polar Decomposition: Ranges

I solved it while writing. Answers still heartly welcome!! :) Given Hilbert spaces. Consider a bounded operator: $$T:\mathcal{H}\to\mathcal{K}:\quad\|T\|<\infty$$ Regard the polar ...
2
votes
3answers
60 views

Why if $T$ is not continuous, then for each $n\in N$, there exists $x_n\in X$ such that $||Tx_n||\ge n||x_n||$

If $X$ $Y$ are Banach space, $T$ is a linear operator between them. I don't understand the following statement: If $T$ is not continuous, then for each $n\in N$, there exists $x_n\in X$ such that ...
1
vote
1answer
34 views

Is the minimiser of the quadratic form of a semi-bounded self-adjoint operator an eigenstate?

I am wondering whether the following fact, for which I know well the proof when $H$ is a Schroedinger operator (see Lieb-Loss, Analysis, Chapter 11), is also true in the general setting used below, ...
1
vote
0answers
32 views

Right shift operator is not compact but is a limit of finite rank operator?

I'm doing a preparation for an exam, and I have a doubt concerning the right shift operator, for example in $l^2$,$S_d : l^2 \to l^2$ such that $(x_1,x_2,\ldots) \mapsto (0,x_1,\ldots)$ is standard to ...
0
votes
0answers
26 views

Matrix representation of an operator

Murphy says : The pure states of $A=K(H)$ are precisely the states $\omega_x : A\to \Bbb C ~~;~~\omega_x(u) = \langle ux,x\rangle $ where $x$ is a unit vector of Hilbert space $H$ . Then he gives ...
0
votes
1answer
19 views

Stieltjes inversion formula

Let $[a,b] \subset \rho(T)$ and $T$ be a self-adjoint operator then I want to show that $0=\frac{1}{\pi} \lim_{\varepsilon \downarrow 0} \lim_{\delta \downarrow 0} \int_{a+\delta}^{b+\delta} ...
3
votes
1answer
52 views

Proof of the spectral theorem

I am currently going to through my proof of the spectral theorem that we had in class, but I feel that I have copied some nonsense from the board. So we defined the Cayley transform $U= ...
2
votes
1answer
24 views

Measure in spectral theorem always positive?

In my functional analysis lecture we introduced the continuous functional calculus on $\sigma(T)$ if $T$ is a self-adjoint operator. Then the Riesz representation theorem gives us that ...
1
vote
1answer
44 views

$\langle Tx,x \rangle =0$ then $T=0$

Given a complex Hilbert space $H$, we have that $\langle Tx,x \rangle =0$ then $T=0$ holds. I looked to some old threads and all of them talked about this by referring to the polarization identity, ...
1
vote
0answers
26 views

An integral identity in the spectral theorem

I have the following question concerning the spectral theorem: Let's suppose that $T$ is some self-adjoint operator (unbounded) in some Hilbert space $H$. Suppose in addition that ...
0
votes
1answer
24 views

Do I have to show this map is well-defined?

Let $H$ be a Hilbert space and $u \in B(H)$. Write $$ H = \overline{\mathrm{im}(u)} \oplus \overline{\mathrm{im}(u)}^\bot$$ and define $v(h) = v(|u|x \oplus z):= u(x)$. Do I have to prove that ...
3
votes
2answers
48 views

Approximate Identities

Statement : Let $U$ be a C* algebra and$\lambda=\left\{A\in U:A\geq 0, ||A||<1\right\}$. If $B\in \lambda$ then if $X\in U$ $$||X^*(I-B)^2X||\leq ||X^*(I-B)X||$$ For reference this is from ...
0
votes
0answers
22 views

Discrete Laplace: ONB

Before, consider the discrete Laplace without boundary: $$\Delta:\ell^2(\mathbb{N})\to\ell^2(\mathbb{N}):(\Delta u)_k:=\frac12(u_{k-1}+u_{k+1})$$ Regard the unitary transformation: ...
2
votes
2answers
33 views

Properties of the multiplication operator, self-ajointness

Let $(\Omega, \Sigma,\mu)$ a measurable space, $f:\Omega\to \mathbb{R}$ $\mu$-measurable. a.My first question: What does "f $\mu$-measurable" mean?I only know, what it means that "f is measurable" but ...
0
votes
1answer
25 views

Continuity of operators

Let $(T_t)_{t \ge0}$ be a family of operators(not necessarily bounded, but all defined on the same domain) and now we have the property $$t \rightarrow 0^+ \Rightarrow ||T_t^2 -T_0^2|| \rightarrow ...
3
votes
2answers
100 views

Are these linear maps bounded?

Let $\mathcal{C}^{\infty}_c$ be the complex vector space of $\mathcal{C}^{\infty}$ functions with compact support in $(0,1)$.Define two norms on it , $\|x(t)\|_u=\text{max}_{t\in (0,1)} \ |x(t)|$ and ...
1
vote
2answers
25 views

Norm of orthogonal projection

Consider $\Bbb R^n$ with the standard inner product and let $P$ be an orthogonal projection defined on $\Bbb R^n$. It is known that the operator norm of $P$ induced by the inner product is less than ...
0
votes
0answers
32 views

null power element in a C*-algebra

Let $A$ be a C*-algebra. Show that there is $x\in A$ such that $x^2=0$. I think in abelian C*-algebra $x^2=0$ if and only if $x=0$(because these elements are continuous functions) Also in certain ...
7
votes
0answers
69 views

Essential Selfadjointness of Quantum Harmonic Oscillator Hamiltonian

The Hamiltonian for the Quantum Harmonic Oscillator is (disregarding constants) the Hermite operator $$ Hf = -f''+x^{2}f, $$ where $\mathcal{D}(H)$ consists of all twice absolutely ...
4
votes
0answers
31 views

$C_0$ semigroup

I was wondering about this: Let $T$ be a self-adjoint operator, then we have $$T = \int_{\mathbb{R}} \lambda \, dE.$$ Does $\int_{\mathbb{R}}e^{it\lambda} \, dE$ then define the $C_0$ semigroup of ...
2
votes
2answers
38 views

Can someone explain the notion of “unbounded” operator as simple as possible?

I've read about these operators in quantum mechanics, but I have never seen them in action. I think that is because I absolutely do not intuitively understand this concept. I've read some stuff online ...
1
vote
2answers
15 views

Exponent of an operator - Existence/Uniqueness?

I have the following questions: When I can define an Expression $A^p$ with an Operator $A$ and a fractional Exponent $p$? Is the root (or fractional or even real exponent) existing for arbitrary ...
2
votes
1answer
23 views

Multiplication Operator and Supremum Norm

Let $m\in C[a,b]$. Consider on $(C[a,b], \|\cdot \|_{\infty})$ the multiplication operator $A: C[a,b] \to C[a,b], \quad Af = mf$. Prove that $\|A\| = \|m\|_{\infty}$. In my book, we are given the ...
1
vote
1answer
30 views

Linear Operator bounded on a basis

Given a Hilbert space $\mathcal H$, a basis $\{e_j\}$ and an injective function $T$ from $\{e_j\}$ to $\mathcal H$ such that $\| T(e_j) \| \leq C$ for all $j$. Can we always extend $T$ to a bounded ...
2
votes
1answer
29 views

positive element in a Banach $*$- algebra

By definition, $a$ is positive in C*-algebra $A$ if $\sigma(a) \subset \Bbb R^+$. I would like to know the definition of a positive element in a Banach $*$-algebra. I think it's the same as the ...
2
votes
0answers
31 views

Solution of non-linear Fredholm(Hammerstein) equation with non-degenerate kernel and reciprocal non-linearity

I have asked this question but got no response. I rephrase it so that anyone who knows operator theory and integral equations would help me out.....I faced a problem in physics which is a non-linear ...
0
votes
0answers
35 views

What is the definition of hyperstonean space?

I've seen several questions and answers on the Gelfand transform for commutative $C^*$-algebras leading to a characterization of commutative Von Neumann algebras as those whose spectrum is ...
0
votes
0answers
18 views

Just what is the importance of operators that produces an eigenvalue?

For some operators, there is a well known eigenvalue associated with it, for example the energy operator in quantum mechanics $i\hbar \partial_t$, this is very important indeed and gives us physical ...
-1
votes
2answers
43 views

Selfadjoint Operators: Characterization

Given a Hilbert space. Symmetric operators can be described by $$\overline{\mathcal{D}(A)}=\mathcal{H}:\quad A\subseteq A^*\iff\langle ...
1
vote
0answers
22 views

Positive linear functional on a C*-algebra is bounded

The following is a theorem of Murphy's C*-algebras and operator theory: My question: I think in the proof of theorem, Murphy uses the assumption $|\tau(a)|<M$ for positive elements $a\in ...
1
vote
1answer
41 views

Resolvent: Norm

Given a Banach space. Consider a closed operator: $$T:\mathcal{D}(T)\to E:\quad T=\overline{T}$$ Due to the Neumann series it holds: $$R(\lambda):=(\lambda- ...
0
votes
1answer
16 views

Show that a sum of operators is bounded.

Let $T$ be an operator for wich there existe $M\geq 0$ such that : $$ \|\frac{1}{n}\sum_{k=0}^{n-1}T^k\|\leq M , \, \forall n\geq 1.$$ Show that for every $r$, $0<r<1$, $$ \|(1-r)\sum _{k\geq ...