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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Question about compact operators

I would like to prove the following, Let $X$,$Y$ be infinite dimensional Banach-Spaces and $T$ a compact, linear and bounded operator. Then there exists a sequence $(x_n)_{n\in\mathbb N}$ with ...
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52 views

Positive operators in Hilbert spaces

Let $H$ be a Hilbert space. I am just asking if there's some reference which studies operators $A$ with this property: $$\left\langle Ax,x\right\rangle \geq0,$$ for all $x\in H$. And $Ax=0$ whenever ...
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34 views

how can i prove this if it is true?

Let Z be a central projection in a von neumann algebra A and Q is a finite projection in A. is it true that ZQ is also finite? if yes how can i prove that? thanks for your help.
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46 views

How to find the spectrum $\sigma_p(P)$

How to find the spectrum $\sigma_p(P)$: Let $P:H\rightarrow H$ be an orthoprojection, $P\neq 0, P\neq I$. could you please help
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41 views

Prove property of adjoint: $(\mathcal{A}^{-1})^*=(\mathcal{A}^*)^{-1}$.

I'm trying to prove it like any other property of adjoint. So, I need to prove following equality: $(\mathcal{A}^{-1}x, y)=(x, \mathcal({A}^{-1})^*y)$. I know it's very basic, but how to prove this ...
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2answers
68 views

How to show that the Volterra operator is not normal

How to show that the Volterra operator: $$V:L_2(0,1)\rightarrow L_2(0,1): x\mapsto \int^t_0 x(s) \, ds$$ is not normal. $t\in (0,1)$ Could you please help with this question.
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2answers
73 views

How to show: $A_y$ has no eigenvectors if $y$ is not constant on any subinterval of $[0,1]$

Let $y\in C[0,1]$ and $A_y : C[0,1]\rightarrow C[0,1]: x\mapsto xy$ How to show: $A_y$ has no eigenvectors if $y$ is not constant on any subinterval of $[0,1]$. Could you please help.
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223 views

Injectivity of the operator $(Ax)(t)=\int_0 ^1 k(s,t) x(s)ds$

Let $X=C([0,1],\mathbb{R})$ (equipped with the supremum norm). Let $A$ be the operator defined for each $x\in X$ by $$(Ax)(t)=\int_0 ^1 k(s,t) x(s)ds,$$ where $k:[0,1]\times [0,1]\to \mathbb{R} $ is ...
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47 views

About the Volterra operator and the approximation property

I need some help with these questions. $\bullet\;$ First of all, if we define the Volterra operator $V:L^{1}[0,2\pi]\rightarrow L^{1}[0,2\pi]$ as $(Vf)(x)=\int_0^xf(t)dt$, Is this operator compact? ...
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38 views

Formula for trace of particular operators

Let $\mathcal{H}$ be the Hilbert space $L^2(\mathbb{R})$. View the Fourier transform as a unitary operator $\mathcal{F} \in B(\mathcal{H})$. For each function $f \in C_0(\mathbb{R})$, let $T(f) \in ...
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2answers
122 views

Compact operators, injectivity and closed range

Let $X$ be a an infinite dimensional Banach space. $A\in B(X)$ is a compact operator. If its range $Im(A)$ is closed in $X$ then $A$ cannot be injective because $A:X\to Im(A)$ would be a compact ...
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3answers
61 views

$\det A \neq 0$. Prove that $\det A^* \neq 0$.

$A$ is matrix representing operator $\mathcal{A}$. $*$ is such operator that respects following equality: $(\mathcal{A}x,y)=(x, \mathcal{A}^*y)$; (I don't know what term is used in English). ...
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1answer
30 views

Prove that $\mathcal{AB}$ is linear operator if $\mathcal{A}$ and $\mathcal{B}$ are linear operators.

It is fairly easy to determine whether $\mathcal{AB}$ is linear when we know $\mathcal{A}$ and $\mathcal{B}$ (for example, $\mathcal{Ax}=(2x_1, 3x_2-x_1)$ and $\mathcal{B}$ is something similar). But ...
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1answer
40 views

Differentiable Path of Operators and their Inverses

Let $\mathcal{H}$ be a separable Hilbert space. Consider a differentiable map $\mathbb{R} \rightarrow \mathcal{B}(\mathcal{H}), t \mapsto A(t)$, where $\mathcal{B}(\mathcal{H})$ is the space of ...
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1answer
150 views

Why study Bergman Spaces?

I'm interested in Operator Algebras and mathematical physics; recently, a friend showed me Duren and Schuster's "Bergman Spaces". I've read about two chapters now and I see there is a nice play ...
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1answer
71 views

Linear and monotone mapping

Let $f:\mathbb{R}^n \rightarrow \mathbb{R}^n$ be continuous and monotone, i.e., $$ \left( f(x) - f(y) \right)^\top \left( x-y\right) \geq 0$$ for all $x,y \in \mathbb{R}^n$. Say for which matrices $A ...
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2answers
27 views

help me please about adjoint of operators in L1

A : L₁→L₁ 1) A x=( x₁, x₂,.....xn , 0,0,....) 2) A x= (λ₁ x₁ ,λ₂ x₂,.....) |λ n|≤1 and λ n ∈ R I need to find adjoint of operators A in given space. ...
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1answer
30 views

Prove that operator of mirror plane $x+z=0$ is linear and find its' matrix.

I am not familiar with term mirror plane , hence I don't know how to solve this problem. As for operator itself, maybe if I select basis $(x,0,0), (0,y,0), (0,0,z)$ then I would express $x+z$ this ...
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33 views

help,example about disjoint operators

$T\colon L^2[0,1]→L^2[0,1]$ is given by $$ Tx(t)=∫_0^1 tx(s)\,ds $$ How can we find adjoint operator of $T$ in this space? $\langle Tx,y\rangle= \langle x,T^*y\rangle$ should be okay.But what ...
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Is there any relation between positive definite operator and an operator that satisfies maximum principle?

Suppose $L$ is a self adjoint differential operator which satisfies maximum principle. Maximum principle: Assume that $u(x)$ satisfies $u(0)\geq 0$ and $u(1)\geq 0$. Now $L$ is said to satisfy ...
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1answer
55 views

Am I wrong ? (2)

Let $X=C[0,1]$ be the space of real continous functions on $[0,1]$. $X$ is a Banach space with the two norms $$|f|_\infty=\sup_{s\in[0,1]}|f(s)|$$ and ...
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1answer
42 views

$(X,|.|_A)$ is Banach implies $A$ is closed

Let $(X,|.|)$ be a Banach space. We know that if $A:X\to X$ is a closed operator then $(X,|.|_A)$ is a Banach space, where $|.|_A$ is the norm defined by $$|x|_A=|x|+|Ax|$$ Then using the "continuity ...
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51 views

Where am I wrong ??

Let $(X,|.|)$ be a Banach space. $A\in B(X)$ a bounded injective operator. Then we can define another norm on $X$ by $$|x|_A=|Ax|.$$ Since we have $$|x|_A\leq |A||x|$$ Then by the result of continuity ...
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0answers
30 views

Application of operator theory in ODE and PDE

I am looking for references of applications of operator theory (especially spectral theory) in ODE, PDE and possibly SDE. I have learnt operator theory in the general set up, but only know little ...
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1answer
21 views

Normal bounded operator

Let $T$ be a bounded normal operator on a Hilbert space. Now I have to show that $T$ is self-adjoint if and only if $\sigma(T) \subset \mathbb{R}$. I already know that for an Abelian unital ...
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1answer
51 views

Compact surjective non injective operator

Let $X$ be an infinite dimensional Banach space. I know that every compact operator $A$ is not bijective or $0\in\sigma(A)$. Fox example the compact operator $A$ defined on $X=C([0,1],\mathbb{R})$ ...
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1answer
40 views

Spectrum of an operator

Let $X=C([0,1],\mathbb{R})$ the Banach space of continuous real functions in $[0,1]$ equipped with the supremum norm. We define the operator $A$ for each $x\in X$ by $$(Ax)(t)=\int_0 ^t x(s)ds, \ \ \ ...
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is $T$ compact operator?

is $T$ compact operator? $T:C[0,1]\rightarrow C[0,1]$: $x(t)\mapsto x(t^2)$ where $t\in[0,1]$ with supremum norm Could you please help.
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1answer
73 views

Frechet/Gateaux differentiability of an integral operator L^2 --> R

Let $f: R \rightarrow R$ be a continuously differentiable function on the real numbers (if needed also infinitely many often differentiable). Define the Operator $F : L^2([0,1]) \rightarrow R$ for $x ...
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1answer
128 views

Fredholm operator norm

I have seen here, that the operator norm of a Fredholm operator $T_k(f)(s):=\int_0^1 k(s,t) f(t) dt $, where $k \in L^2([0,1]^2)$ and $f \in L^2([0,1])$ is not equal to the $L^2$ norm of the Kernel. ...
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44 views

I have to decide whether an operator is closed

So here is my problem, I have to decide whether the following operator is closed, $$\frac{\mathrm{d}}{\mathrm{d}x}:C^2([0,1])\subset C^0([0,1])\rightarrow C^0([0,1])$$ with the $||\cdot||_{\infty}$ ...
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17 views

Prove that $A\int_0^\infty S(t) u dt=\int_0^\infty S(t) A u dt$ if A is a closed operator

From Wikipedia: Closed linear operators are a class of linear operators on Banach spaces. They are more general than bounded operators, and therefore not necessarily continuous, but they still ...
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1answer
22 views

Proof of Strong Operator Convergence Theorem

Recall the theorem : $T_n \in B(X,Y)$ where $X,\ Y$ are Banachs, is strongly convergent iff (a) $ \parallel T_n \parallel $ is bounded (b) $T_nx$ is Cauchy where $x$ is in total subset ...
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1answer
68 views

Proving that $AB-BA=cI$ for nontrivial $c \in \mathbb{C}$

I have a homework question I can`t solve: Let $X$ be a normed linear space, $A,B \in B(X)$. Show that there exists no nontrivial $c \in \mathbb{C} $ such that $AB-BA=cI$. Thanks alot already guys! I ...
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Doubt on eigenvalues of normal operators

I'm trying to understand the solution of the following problem: $T$ is a normal operator. If $T( v)=\lambda v$, then $T^*(v)=\bar\lambda v$: The solution is: I didn't understand why we ...
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Holomorphic Functional Calculus

Framework: Consider a Banach space: $$(E,\|\cdot\|)$$ Given an unbounded operator: $$T:\mathcal{D}(T)\to E\qquad\mathcal{D}(T)\subseteq E$$ together with its resolvent map: ...
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1answer
69 views

How to find if it is a compact operator

How to find if it is a compact operator: $F\colon C[0,1]\rightarrow C[0,1]$ : $x(t)\mapsto \int^1_0 \cos(t^2+s^2)x(s)ds$ Could you please help with this question.
3
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1answer
60 views

Want to show that an operator is not surjective

So here is my problem, Let $$M_1:L^1\rightarrow L^1$$ $$f(x)\mapsto \arctan(x)f(x)$$ In order to compute the spectrum of $M_1$ I am investigating for which $\lambda\in\mathbb C$ the following map is ...
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1answer
24 views

Injectivity and surjectivity of $\lambda I-A$.

Let us $A$ a square matrix, $\lambda\in \mathbb R^+$, $I$ identity matrix, R a operator, X Banach space. If $$(\lambda I-A) Ru=u \ \ (u\in X)$$ and $$R(\lambda I-A) u=u \ \ (u\in X)$$ then can we ...
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1answer
63 views

Exponentiation of imaginary operator

It is very easy to prove that if $D=\dfrac{d}{dx}$, then $(e^{nD}f)(x)=f(x+n)$ about $x=m$ in the real numbers. Proof: $$(e^{mD}f)=\sum^\infty_{n=0}\dfrac{D^nf}{n!}m^n\\ \implies ...
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Understanding the definition of a Integral

Definition: Let $X$ be a Banach space and $I$ the identity operator on $X$. A family $\{T(t)\}_{t\geq 0}$ of bounded linear operators from $X$ into $X$ is a semigroup of bounded linear operator on ...
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derivation of divergence from nabla operator

For a two dimensional orthogonal curvilinear coordinate system $(t_1, t_2)$, we have the position vector $r$, where $h_i = | \frac{\partial r}{\partial t_i} |$ are the scale factors and $a_i$ are the ...
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45 views

Polynomial Calculus on Spectrum: well defined?

Consider a bounded operator over a Banach space: $T\in\mathcal{B}(E)$ Apply polynomial calculus on the the chosen operator: $p(T),p\in\mathbb{C}[X]$ Why do we need to prove that when two polynomials ...
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0answers
29 views

Cauchy Schwarz inequality with an operator

The standard Cauchy-Schwarz inequality is given by, $|\langle\Phi|\Psi\rangle|^2\le\langle\Phi|\Phi\rangle\langle\Psi|\Psi\rangle$ But now I'm intressted in what happens to ...
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55 views

Root of polynomial implies vanishing remainder. Application to spectral theory!

Framework: Consider a unital ring: $e\in R$ and a given polynomial: $p\in R[X]$ (Note that I do not require the ring to be an integral domain.) Problem: If it has a root then it factorizes: ...
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1answer
33 views

Hermitian and Diagonal Matrix Norm inequality

I have a matrix inequality that I think is true, but I can't prove. $D_1$ and $D_2$ are diagonal matrices with non-negative entries. $M_1$ and $M_2$ are positive definite matrices. I want to show ...
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1answer
53 views

Question about finite rank operators

Let $X$ be a normed space, $\mathcal{F}(X)$ the algebra of all operators on $X$ with finite fank, then $\mathcal{F}(X)$ is the unique minimal ideal of $\mathcal{K}(X)$ the algebra of all compact ...
12
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1answer
206 views

A property of exponential of operators

Let $X$ be a Banach space. $A\in B(X)$ is a bounded operator. we can define $e^{tA}$ by $$e^{tA}=\sum_{k=0}^{+\infty}\frac{t^kA^k}{k!}$$ I am interested in this property: If $x\in X$, such that the ...
2
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2answers
55 views

Is this operator bounded ??

Let $X$ be the Banach space $X:=\{ f\in C(\mathbb{R},\mathbb{R}),\sup_{t\in \mathbb{R}}|e^{-s^2}f(s)|<+\infty \}$ equipped with the norm $$|f|_X=\sup_{t\in \mathbb{R}}|e^{-s^2}f(s)|$$ I want to ...
3
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2answers
53 views

an invariant of $C^{*}$ algebras

consider the following property (invariant) for complex $C^{*}$ algebras: "$T(x)=x^{*}$ is the only non zero $\mathbb{R}$-linear map on $A$ which satisfies $T(x)T(y)=T(yx)$." Questions: 1)Some ...