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.

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Semigroups: Product Rule [closed]

Given a Banach space $E$. Consider C0-semigroups: $$S,T:\mathbb{R}_+\to\mathcal{B}(E)$$ Then the product rule holds: $$(TS)'(t)x=T'(t)S(t)x+T(t)S'(t)x$$ How to prove this from scratch?
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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 ...
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35 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 ...
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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 ...
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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 = ...
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Compute the spectrum of the integral operator $K:L^2([0,1]) \to L^2([0,1])$ defined as $(Kx)(t) = \int_0^t x(s) ds$

Let $K:L^2([0,1])\rightarrow L^2([0,1])$ be the linear operator defined by $$(Kx)(t)=\int_0^tx(s)ds, \quad x \in L^2([0,1]).$$ Now I have to compute the spectrum, but I don't have any idea how to do ...
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119 views

Integration by parts to find 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 ...
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88 views

Modulus: Invariant Domain

Given Hilbert spaces $\mathcal{H}$ and $\mathcal{K}$. Consider an operator: $$M:\mathcal{H}\to\mathcal{K}:\quad \|M\|=1$$ Regard dense domains: ...
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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 ...
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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 ...
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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 $p= \frac{1}{2}(b+c)$ ...
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34 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 ...
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Spectral Measures: Helffer-Sjöstrand

Given a Hilbert space $\mathcal{H}$. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ Regard a function: $$f\in\mathcal{C}^\infty_0(\mathbb{R}):\quad ...
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101 views

Spectral Measures: Stone's Formula

Given a Hilbert space. Consider a Hamiltonian: $$H:\mathcal{D}(H)\to\mathcal{H}:\quad H=H^*$$ Regard its spectral measure: $$E:\mathcal{B}(\mathbb{R})\to\mathcal{B}(\mathcal{H}):\quad ...
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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 ...
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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 ...
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121 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 ...
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36 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 ...
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Polar Decomposition: Ranges

This is just a note. Given Hilbert spaces $\mathcal{H}$, $\mathcal{K}$. Consider a closed operator: $$A:\mathcal{D}(A)\to\mathcal{K}:\quad A=A^{**}$$ Construct its modulus: ...
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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 ...
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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, ...
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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 ...
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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 ...
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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} ...
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87 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= ...
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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 ...
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$\langle Tx,x \rangle =0$ 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, ...
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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 ...
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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 ...
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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: ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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53 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 ...
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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 ...
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70 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- ...
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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 ...
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81 views

Intuitive understanding of the operator norm?

I understand various vector norms, but I don't understand operator norms. Specifically, norms on linear operators. Can anyone explain them?
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Pure Math Research into Operator Fields

Has any work been done on operator fields in the pure math world? They are a big piece of quantum field theory, but I can't find anything about them outside of that messy subject. Of course, I mean ...