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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Intuition behind: Integral operator as generalization of matrix multiplication

So I am teaching myself more in-depth about integral operators and every once and awhile I see this little 'factoid', that integral operators are generalizations of matrix multiplications. In ...
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1answer
56 views

Closure of an Operator in $l^2$

Let $l^2$ denote the Hilbert space of all complex sequences $\phi = (\phi_j)_{j=0}^{\infty}$ such that $\sum_{j=0}^{\infty} |\phi_j|^2 < \infty$. Consider the linear subspace of $l^2$ defined by ...
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A relation between two properties of sequences of operators

We have $(T_l)_l$ a sequence of bounded linear operators from $\ell^2$ to $\ell^2$. $\bullet$ We say $(T_l)_l$ satisfies the property "A" if $\sup_{||x||=1}\sum_{l=1}^\infty||T_l(x)||^2<\infty$. ...
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Idempotents which are not Mouray von neumann equivalent to its adjoint

What is an example of a $C^{*}$ algebra with an idempotent $e$ such that $e$ is not Mourray Von neumann equivalent to $e^{*}$?
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28 views

Algebra of Linear differential operators, question on Commutativity and Association

The following is a discussion on the following second differential equation $$ \frac{dy^2}{dx} - y = 0 $$ So, let us introduce the following, convention and definition, represent the derivative ...
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1answer
21 views

the c*-algebra generated by the Volterra operator

Let V be the Volterra operator on $\mathscr{L^2(0,1)}$.$V(f)(x)=\int_{0}^{x}{f(y)dy}$. Show that $C^*(V)$, the smallest C* algebra generated with V, is $\mathbb{C}+\mathscr{B_0(L^2(0,1))}$ where ...
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0answers
23 views

Distance from image of bounded operator

Let $A:H\rightarrow H$ be a bounded linear operator on Hilbert space $H$. Suppose we have $x\in H$ and $r>\mathrm{dist}(x,A(H))=\inf\{\|x-Ah\|,\ h\in H\}$. How to prove that then there exist ...
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5answers
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How could we define the factorial of a matrix?

Suppose I have a square matrix $\mathsf{A}$ with $\det \mathsf{A}\neq 0$. How could we define the following operation? $$\mathsf{A}!$$ Maybe we could make some simple example, admitted it makes any ...
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3answers
36 views

Spectral radius and dense subspace

Let $X$ be a Banach space, and let $E$ be a dense subspace of $X$. Let $A: X \to X$ be a bounded operator on $X$ that maps $E$ to itself. Assume that the spectral radius of $A$ restricted to $E$ is ...
3
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2answers
32 views

Composition involving bounded linear operators

I recently come across the following statement mentioned in a proof: Let $X,Y$ be normed linear spaces and $T:X \rightarrow Y$ be a linear operator. if for every bounded linear functional $U: Y ...
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Differential equations and Fourier and Laplace transforms

Why do both the Fourier transform and the Laplace transform appear in the study of differential equations? I've never understood why there are some situations where the Fourier transform is used and ...
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71 views
+50

Complex Root of Unity Analogue of Forward Difference Operator

In my studies I have come across a couple of operators; in particular; $$\Delta[f(x)]=f(x+1)-f(x)$$ $$\Delta^*[f(x)]=f(x+1)+f(x)$$ $\Delta$ has been called the Forward Difference Operator. I was ...
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1answer
12 views

infimum of operator norms of iterations of linear operators

I am currently reading a proof in which a fact is used without proof: For a Banach space $X$ and a bounded linear operator $T: X \to X$, $$ \lim_{n \to \infty} \| T^n \|^{\frac{1}{n}} = \inf_{n ...
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0answers
33 views

example of positive operators a,b, $a\le b$ but $b^2-a^2$ is not positive [on hold]

Give an example of a C*-algebra $\mathscr{A}$ and positive elements a,b in $\mathscr{A}$ such that $a\le b$ but $b^2-a^2\notin \mathscr{A}_+$, i.e. $b^2-a^2$ is not positive element in $\mathscr{A} $
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221 views

Factorial of a matrix: what could be the use of it?

Recently on this site, the question was raised how we might define the factorial operation $\mathsf{A}!$ on a square matrix $\mathsf{A}$. The answer, perhaps unsurprisingly, involves the Gamma ...
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2answers
39 views

Matrix of linear operator in different bases [PMA Rudin]

$\mathbf{9.35}\quad$ Theorem $\ \; $ If $[A]$ and $[B]$ are $n$ by $n$ matrices, then $$\det([B][A])=\det[B]\det[A].$$ $\mathbf{9.36}\quad$ Theorem $\ \; $ A linear operator $A$ on $R^n$ is ...
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14 views

the c*-algebra generated by a closed ideal and a c*-subalgebra

If $\mathscr{A}$ is an unital c*-algebra, $I$ is a closed ideal of $\mathscr{A}$ ,and $\mathscr{B}$ is a unital c*-subalgebra of $\mathscr{A}$ . Show that the c*-algebra generated by ...
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1answer
26 views

Support of a normal pure state is a rank one projection

Let $\phi$ be a normal pure state on a w*-algebra $M$ and $\{\pi, \xi, H\}$ its GNS representation associated to $\phi$. Suppose projection $e$ is the support of $\phi$. Show that $\pi(e)$ is a rank ...
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0answers
30 views

Convolution operator is normal

Consider the convolution operator $$Tf(s)=\frac{1}{2\pi}\int_0^{2\pi}f(t)h(s-t)\,\,dt,\quad f\in L^2[0,2\pi]$$ where $h:\Bbb R\to \Bbb C$ is a $2\pi$-periodic function, square integrable on ...
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1answer
249 views

Comparison of Strong OPerator and Weak * Topologies on B(H)

It is known that in $\mathfrak{B}(\mathbb{H})$, the weak operator topology (WOT) is contained in both the strong operator topology (SOT) and $\sigma$-weak topology. In general the SOT and the ...
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1answer
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Continuous but not compact operator on $L^2(0,\infty)$

Define the following operator on $L^2(0,\infty)$: $$Tf(x)=\frac{1}{x} \int_0^xf(y)dy,\quad f\in L^2(0\infty).$$ I would like to see that it is continuous but not compact. So, this is an integral ...
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2answers
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Self-adjoint operators, projections, and resolutions of the identity.

In my Functional Analysis course, we're discussing the Spectral Theorem and the like. One question from a previous exam states the following: Let $H$ be Hilbert over $\mathbb C$, let $T \in B(H)$ be ...
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1answer
22 views

Closed ideals in $\mathbb B(H)$

Let $\mathbb{H}$ be a non-separable Hilbert space. If $\alpha$ is an countably many infinite cardinal number, let $I_{\alpha}=\{A\in \mathbb{B(H)}\:dim~ cl(ran A)\le \alpha\}.$ Show that ...
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Finite dimensional Banach algebras whose $K_{0}$ group is a non trivial finite group

Motivated by this question we ask Is there a finite dimensional Banach algebra $A$ such that $K_{0}(A)$ is a nontrivial finite group? I understand from the above link and this post that any ...
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1answer
17 views

Positive self-adjoint operators and norm resolvent convergence

I recently came across a reference to the following Theorem (Simon/Reed, Methods of Modern Mathematical Physics, viii.25) and am now trying to figure out a proof for it: If $A_n$ and $A$ are positive ...
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1answer
50 views

Example of operator with spectrum equal to $\mathbb{C}$?

In my Functional Analysis course, we proved that for a (possibly unbounded) operator $T$ that is densely defined, closed, and symmetric, exactly one of the following four occurs: $\sigma(T) = ...
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2answers
477 views

Fourier transform as diagonalization of convolution

I've read this in a lot of places but never quite got how this is true or meant. Let's say we have a convolution Operator $$ A_f(g) = \int f(\tau)g(t-\tau)d\tau $$ and apply it to $g(t)=e^{ikt}$. ...
3
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1answer
43 views

Example of a self-adjoint bounded operator on a Hilbert space with empty point spectrum

Good day, I wanted to find a self-adjoint bounded operator on a Hilbert space with empty point spectrum i.e. $$ T = T^* ~\text{but}~ \sigma_p(T)= \emptyset $$ Some definitions and results of the ...
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How should $A^α$ be defined for real $α ∈ [0,∞)$ and $A\in M_n(\mathbb C)$?

Let $A\in M_n(\mathbb C)$ be arbitrary. I'm interested to know How should $A^{\alpha}$ be defined for real $\alpha\in [0,\infty)$? When $A$ is nonsingular, we can define $A^{\alpha}=\exp(\alpha ...
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1answer
29 views

‎Jointly ‎continuous of product in $B(H)$

‎Let ‎$‎B(H)‎$ ‎be the set of‎ ‎bounded ‎operators ‎on a Hilbert space ‎‎$‎‎H$.‎ ‎ I ‎know that ‎$u_{‎\alpha‎}‎\longrightarrow u‎$‎‎‎ ‎in ‎S.O.T ‎if ‎and ‎only ‎if‎ ‎$u_{‎\alpha‎}(x)‎\longrightarrow ...
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Adjoint of an Operator in $l^2$

Let $l^2$ be the Hilbert space of all complex sequences $\phi =(\phi_j)_{j=0}^{\infty}$ such that $\sum_{j=0}^{\infty} |\phi_j |^2 < \infty$. Set $D= \{ \phi \in l^2 : \sum_{j=0}^{\infty} j ...
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1answer
20 views

Largest invariant subspace

If $A$ is a $n\times n$ matrix with complex entries, denote by $inv(A)$ the dimension of a largest dimensional non-trivial invariant subspace of $A$. What is: $$\inf\{inv(A): ...
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1answer
20 views

Small perturbations and eigenvalues

Suppose $A$ is a $n\times n$ matrix. Given $\epsilon>0$, can one find a rank one matrix $B$ with euclidean norm at most $\epsilon$ such that $A+B$ has $n$ distinct complex eigenvalues? Given a ...
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1answer
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Double dual of the space of bounded operators on Hilbert space [duplicate]

Every Banach space $X$ is canonically, isometrically embedded in its bidual $X^{**}$. But it is not always $1$-complemented in the bidual: for example, there is no projection from $\ell_\infty$ onto ...
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Eigenfunctions of integral operator

I am faced with the problem of calculating the eigenfunctions for an operator of the form: $(Kf)(x) = \int_{-\infty}^{\infty} K(x-\alpha y) f(y)dy $ Does anyone know for which functions (or types of ...
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2answers
50 views

Prove that there is no norm for to make this mapping continuous

I am dealing with an exercise which is as follows: Show that there is no norm such that the set of all the mappings $T_a$ which map every element $f\in C(\mathbb{R}, \mathbb{R})$ (where the latter is ...
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support of an operator on a Hilbert space [closed]

let $x\colon\mathcal{H}\to\mathcal{H}$ be a self-adjoint operator, the support $s(x)$ of $x$ is defined as the smallest projection $e\in B(\mathcal{H})$ such that $ex=xe=x$. let $A=\int\lambda \, ...
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Books for studying Dirac Operators, Atiyah-Singer Index Theorem, Heat Kernels

I am interested in learning about Dirac operators, Heat Kernels and their role in Atiyah-Singer Index Theorem. From various sources (including this very helpful question), I have come to know of ...
2
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1answer
42 views

Square Root of the shift operator indexed by $\mathbb{Z}$

My question is very similar to this question, but instead of indexing by $\mathbb N$ I am indexing by $\mathbb Z$. Consider the shift operator $T : \ell^1(\mathbb Z) \to \ell^1(\mathbb Z) $ given by ...
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There are infinitely many projections

Can anyone explain please why such projections are infinitely many?
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Brief moment from Definition 9.30 from Rudin's PMA

I have some problems with claim which is marked with red line. Claim: Let $A\in L(\mathbb{R}^n)$. Then $A$ is invertible if and only if $\text{rk}A=n$. Proof: $\Rightarrow$ We know that $A\in ...
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K theory of finite dimenional Banach algebras

Is there a reference which studied the K theory of finite dimensional Banach algebras? In particular is there a finite dimensional Banach algebra whose $K_{0}$-group is a non trivial finite group?(I ...
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1answer
24 views

Weak operator topology convergence of hermitian operators

Let $\{A_i\}$ be a net of hermitian operators on a separable Hibert space $\mathbb{H}$ and suppose that there is a hermitian operator T such that $A_{i}\le T$ for all i. If $\{<A_i h,h>\}$ is an ...
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1answer
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Normal Operators: Superalgebra (I)

Given a Hilbert space $\mathcal{H}$. Consider normal operators: $$N:\mathcal{D}(N)\to\mathcal{H}:\quad N^*N=NN^*$$ Denote their calculus: ...
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Show $\Bigl\{\sqrt{2\over {\pi}}\sin (nx)\Bigr\}_{n=1}^{\infty}$ is an orthogonal basis of $L_2[0,\pi]$

Show $\Bigl\{\sqrt{2\over {\pi}}\sin (nx)\Bigr\}_{n=1}^{\infty}$ is an orthogonal basis of $L_2[0,\pi]$. What I need is a verification and guidance. I managed to show that the set is orthogonal. My ...
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35 views

Conditions under which an Convolution operator is normal.

I have a possibly complex valued convolution operator given by $ \int_{\mathbb{R}}K(x-y)f(y)dy$ I know that the operator is self-adjoint if $K(x)=\overline{K(-x)}$ holds. But are there softer ...
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1answer
25 views

Examples of bounded linear operators with range not closed

I've been trying to get some intuition on what it means for a bounded linear operator to have closed range. Can anyone give some simple examples of such an operator that does not have closed range? ...
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properties of vector space

Let $E$ be Banach space and $0<r<1$, $1\leq p<\infty$. Define the set $A$ as follow $$A=\left\{(x_j)_{j\in\mathbb{N}}\subset E :\sum\limits_{j=1}^{\infty}\left[\left|\left\langle ...
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2answers
35 views

Prove equality of norms of operators

Let $e_i$=${\{\delta_{k,i}}\}_{k\ge1}$ $\in$ $l_2$, $i\ge1$, $A_n$ and $B_n$ - operators that are defined like this: $A_n\{x_i\}_{i\ge1}$ = $x_ne_1$, $B_n\{x_i\}_{i\ge1}$ = $x_1e_n$ ...
2
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1answer
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property of the Gelfand transform: why does an isometry map closed sets to closed sets?

The following is a theorem about self-adjoint subalgebra of $C(X)$ where $X$ is compact Hausdorff and the first half of its proof: Here are my questions: Why $\Gamma:\mathfrak{U}\to ...