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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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} Im(\...
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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= (T-i)(T+i)^{-...
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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 $l_x(f):=\...
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$\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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31 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 $...
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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: $$U:\mathcal{L}^2(-...
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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 $$t \rightarrow 0^+ \Rightarrow ||T_t^2 -T_0^2|| \rightarrow 0$$...
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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 C*-...
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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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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 A\varphi,\psi\rangle=\langle\varphi,A\psi\rangle\quad(\varphi,\...
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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 A_{\|.\|\...
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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- T)^{-1}:\quad\|R(\lambda)\|\geq\frac{1}{d(...
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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 0}...
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135 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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50 views

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 "...
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Bounded Linear Operator and the Adjoint

Let $S$ be a linear operator with dense domain $\mathcal{D}(S)$ in the Hilbert space $\mathcal{H}$. Assume that the domain $\mathcal{D}(S)$ belongs to a larger domain, namely $\mathcal{D}(S) \subset \...
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Eigenvalues of an operator induced in a quotient space

Give an example of a vector space $V$, an operator $T \in \mathcal L(V)$ and a $T$-$\space$invariant subspace $U$ of $V$ such that $T/U$ has an eigenvalue that is not an eigenvalue of $T$. Attempt: I ...
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Minimal polynomial in infinite dimension

Let $T$ be a operator on a complex Banach space $E$. Show that there exists a polynomial $P$ such that $P(T)=0$ if and only if the spectrum of $T$ consists in a finite number of eigenvalues. Firt ...
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Maximal Ideal Spaces

STATEMENT: Let $C_b(\mathbb{R})$ be the $C^*$-algebra of bounded continuous functions on $\mathbb{R}$. Let $A$ be the $C^*$-subalgebra of $C_b(\mathbb{R})$ generated by $C_∞(\mathbb{R})$ together with ...
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On the sum of projection operators

It is known that a projection operator can be written explicitly as follows: $$\hat{P} = \sum_{k=1}^n \hat{P_k} = \sum_{k=1}^n | k \rangle\langle k|$$ where $\{|k\rangle$, $k= 1,\ldots,n\}$ are the ...
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Conjugate of a hermitian operator also hermitian

I want to prove that if $D$ is a hermitian operator, then $D^*$ is also a hermitian operator. $D$ is a hermitian operator implies that $(f,Dg) = (Df,g)$ where $f$ and $g$ are functions. Therefore, I ...
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Contraction Mapping Principle

Let $X$ be a Banach space and $T\in\mathscr{L}(X,X)$ with $\|T\|_*<1$. Use the Contraction Mapping Principle to show (where $I$ is the identity map on $X$) that $I-T\in\mathscr{L}(X,X)$ is ...
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Decomposition of resolvent in projections

I am reading the book Perturbation theory for linear operators from Kato. He defines (§5 Section 3) for an operator $T : X\to X$ on a finite Banach Space the resolvent as $$ R(x) = (T- x)^{-1}.$$ ...
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Iterations $F^n_h[f]$ of the operator $F_h[f]=D_h[f]\circ f^{-1}$

Let the $H$ be a collection of real valued invertible functions, define $f\circ g$ as composition, $f+g$ as the function $f+g(x):=f(x)+g(x)$ and define a family of functions $\{D_h\}_{h\in \Bbb R}:H\...
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Laurent Series of operator-value function

I am reading the book 'Perturbation Theory for Linear Operators' from Kato. He defines in his Book (Chapter 1 §5) the resolvent for some operator $T: X \to X$ on a finite Banach space $X$ as a ...
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How is the following expresson be obtained and the meaning of the expression in blue box?

Let me introduce the term {$E_\lambda:\lambda \geq0$} is the spectral resolution of identity of a self adjoint densely defined, positive and closed operator $A:D(A)\subset X\rightarrow X$ , Where X be ...
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Compact Operators: Weak Convergence [duplicate]

Problem Given Banach spaces $X$ and $Y$. Consider a compact operator $C\in\mathcal{C}(X,Y)$. Then weak convergence is turned into strong convergence: $$x_n\rightharpoonup x\implies Cx_n\to Cx$$ I'd ...
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Question about the Image of a compact transformation of a Hilbert space

$T$ is a compact operator on a Hilbert space. Show that $\operatorname{im}(T)$ does not contain a closed infinite dimensional subspace. Here is my attempt at the problem: Suppose that $\operatorname{...
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Existence of compactly supported Fourier transforms on LCA groups

I'm trying to prove the following theorem: The following are equivalent for a locally compact abelian group: $G$ has an open compact subgroup. There exists a nonzero $f\in C_c(G)$ such that $\hat{f}\...
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Compact Approximation

This is meant as lemma for: Approximation Property Given a Banach space $E$. Denote compact operators by $\mathcal{C}(E)$. Consider a compact domain $C\subseteq E$. Then there is a compact ...
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Spectral mapping theorem of the measurable functional calculus

Let $H$ be a hilbert space and $T\in L(H)$ a self adjoint operator. Show that we have in general $\sigma(f(T))\neq f(\sigma(T))$ Any tips? If I choose a self adjoint operator how the measurable ...
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Compact operator as certain limit

Let $H$ be an infinite-dimensional Hilbert space with basis $\{e_i\}_{i=1}^\infty$. Let $P_n := \sum_{i=1}^n e_ie_i^*$, i.e. $P_n$ is the projection onto the span of the first $n$ basis vectors. Let $...
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442 views

Inverse of laplacian operator

I recently read a paper, the author treats $$\int_{\mathbb{R}^d}f(y)\cdot \frac{1}{|x-y|^{d-2}}\,dx = (- \Delta)^{-1} f(y)$$ up to a constant in $\mathbb{R}^d$. I am not familiar with unbounded ...
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Tensor Product: Boundedness

This thread is just a note. Given Hilbert spaces. Then boundedness will be inherited: $$A,B\text{ bounded}\implies A\otimes B\text{ bounded}$$ Especially, the bounds multiply: $$\|A\otimes B\|=\|A\|\...
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Application of the spectral mapping theorem

Let $T:L^2((0,2)\rightarrow L^2((0,2))$, $(Tx)(t):=\begin{cases} x(t+1), & 0<t<1\\ 0,& \text{elsewhere} \end{cases} $ Show that $T$ is well defined and $\sigma(T)=\sigma_p(T)=\{0\}$ ...
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Spectrum of Laplacian on Half line. $\left [0, \infty \right)$

I would like to calculate the spectrum of Dirichlet and Neumann Laplacian of the domain $\left [0,\infty \right)$. To be precise, Define the Operator $T$ on $L^2\left[0,\infty\right)$ as $Tf=-f''$ ...
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Tensor Product: Closability

This was a real question of mine. Given Hilbert spaces. Then closability will be inherited on tensor products: $$A,B\text{ closable}\implies A\otimes B\text{ closable}$$ For simple tensors this is ...
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An invertible hermitian element of a C*-algebra has a logarithm

Suppose $ A$ is a C*-algebra. Show that an invertible hermitian element of $A$ has a logarithm. ($a$ has a logarithm if there is an element $b\in A$ such that $e^b=a$) If $a\in A_+$ then it's easy ...