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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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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2answers
56 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 ...
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0answers
41 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
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99 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 ...
2
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2answers
42 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 ...
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17 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
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1answer
47 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 ...
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1answer
38 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
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1answer
40 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
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0answers
65 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 ...
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0answers
56 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 ...
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29 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 ...
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52 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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0answers
42 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 ...
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1answer
66 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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1answer
17 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 ...
2
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1answer
37 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 ...
2
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2answers
41 views

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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180 views

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 ...
0
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1answer
20 views

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 ...
3
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2answers
130 views

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 ...
2
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1answer
150 views

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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3answers
23 views

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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3answers
60 views

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 ...
2
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1answer
30 views

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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34 views

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 ...
0
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1answer
60 views

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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1answer
42 views

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 ...
6
votes
1answer
112 views

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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1answer
39 views

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 ...
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0answers
10 views

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 ...
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1answer
40 views

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 ...
3
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1answer
65 views

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 ...
2
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1answer
40 views

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 ...
0
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1answer
64 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 ...
0
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1answer
25 views

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 ...
4
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1answer
44 views

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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1answer
89 views

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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1answer
25 views

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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45 views

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 ...
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54 views

is the pullback operator associated to a flow bounded in L^2?

Let $M$ be a smooth compact manifold with a finite Borel measure $m$. Let $\{f_t\}_{t\in\mathbb R}$ be a $C^1$ flow on $M$. That is, a $C^1$ function $$ \mathbb R\times M\ni(t,x)\mapsto f_t(x)\in M $$ ...
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36 views

Norm of Fredholm operator in $L^1$

Let $T:L^1([0,1])\rightarrow L^1([0,1])$ be the Fredholm integral operator given by $$ Tf(x)=\int_0^1 k(x,y)f(y)\, dy $$ where $k \in C([0,1]^2)$ is called the kernel of $T$. My problem is to find ...
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1answer
46 views

Is everything an operator?

For example, I have some number $\alpha$ and a function $f$. Now I multiple this constant $\alpha$ with $f$ and get $\alpha * f$. Now I claim that $\alpha$ is an operator, $f$ my eigenvector, with ...
4
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2answers
32 views

If the scalar product are equal then the operators are equal.

I want to show the following: Let H be a $\mathbb C$ -hilbert space and $S,T\in L(X)$ If $\langle Sx,x \rangle = \langle Tx,x \rangle$ for all $x\in H$, then $S=T$ Any hints for me?
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1answer
38 views

Question about the notation $S \subset T$ ,where $S$ and $T$ are operators

I want to prove that if $S\subset T$. Then $T^{*}\subset S^{*}$. But what does $S\subset T$ mean? $S$ and $T$ are operators and not sets.. :/
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18 views

Sums of two closed and closed / continuous operators

Let $X$ be a normed space and $A_j:D(A_j)\rightarrow X$ (j=1,2) linear. (i) If $D(A_1)=X$, $A_1$ continuous and A_2 closed. Do we have $A_1+A_2:D(A_1)\cap D(A_2) \rightarrow X$, $x\mapsto A_1x+A_2x$ ...
2
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1answer
46 views

Unitary Equivalent of Derivative in Fourier Space

It is known that for $L^2(\mathbb R)$ the operator $Tf(x) = if'(x)$ is unitary equivalent to $\hat T \hat f(\xi )= \xi \hat f(\xi) $. Where domain of T is $H^1(\mathbb R)$. Hence the Spectrum of T in ...
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1answer
33 views

Do spectrum and Eigenvalues of $Af=-f''$ concide (under dirichlet boundary conditions)

I am asked to show that for the operator $$ Af = -f'' $$ with $D(A)=\left\{f\in H^2(0,1), f(1)=f(0)=0 \right\} \subset L^2(0,1)$ is self Adjoint in $L^2(0,1)$ (This part is solved). I cannot see ...
3
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2answers
31 views

For positive operators $A$ and $B$ with $A^6=B^6$ show that $A=B$

Since $A$ and $B$ are positive, I managed to show that $A^6$ and $B^6$ are positive. Now, I can use the fact that there exists a unique square root of both of those and since they're equal, their ...
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0answers
48 views

Show that the given family of bounded operators on a hilbert space form a semi group.

Suppose $A:D(A)\subset H\rightarrow H$ is a self adjoint, densly defined closed operator and it is also positive operator i.e $<Au,u>\geq0 $, for all $u\in H$ ,where $H$ is a hilbert space . ...