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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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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30 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
79 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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56 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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29 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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38 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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137 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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70 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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26 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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25 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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28 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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16 views

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
53 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
39 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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48 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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25 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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18 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
39 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
27 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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241 views

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
52 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
105 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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43 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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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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18 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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64 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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21 views

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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63 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.
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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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21 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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57 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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68 views

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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34 views

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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40 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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22 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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1answer
53 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
22 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 ...
2
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1answer
32 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 ...
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194 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 ...
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48 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 ...
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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 ...
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1answer
49 views

Norm of the multiplication operator

Let $f \in L^\infty[0,1].$ It is clear that the norm of the multiplication operator $M_f : g \mapsto fg$ on $L^p[0,1]$ is $\|f\|_\infty.$ What happens in the noncommutative situation? Let us ...
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Verification of a contraction

Let $A\colon \text{dom}(A) \to \mathcal{H}$ be a densely defined symmetric operator on a Hilbert space $\mathcal H$. The symmetry implies that $$ \|(A + i)f\|^2 = \|Af\|^2 + \|f\|^2 \quad \text{for ...
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1answer
24 views

Covolution (space) over compact Lie groups

Let $G$ be a compact Lie group. Is there any way one can characterize the functions $\phi$ of the form $\phi=\psi\ast \psi^\ast$ in $C^\infty(G)$ where $\psi\in C^\infty(G)$? Here as usual ...
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35 views

Is it possible to consider an approximation to a (non-self adjoint) operator with a self adjoint one?

In operator theory it's wonderful if we have a self-adjoint operator (non necessarily bounded) due to all the work that has been done using their symmetry,... etc. I.e there are many powerful tools. ...
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Which operators commute with integration?

Leibniz proved that $D_y\cdot D^{-1}_x=D_{x}^{-1}\cdot D_y$, where $D_x=\frac{\partial}{\partial x}$. It follows that $D^n_y\cdot D^{-1}_x=D_{x}^{-1}\cdot D^n_y$ where $n \in \mathbb{N}.$ I've not ...
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1answer
30 views

Question about the operator norm on $\mathbb R^2$

So here is my question, I have to decide whether the following statement is true Let $T$ be an isomorphism on $\mathbb R^2$. Then $$\|T\|=\frac{1}{\|T^{-1}\|}$$ I am pretty sure that the statement ...
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1answer
52 views

Linear-Bounded Functional Proof

How can I prove that the following functional defined in $C[a,b]$ is both linear and bounded? $$ f: C[a, b] \to \mathbb R, $$ $$ f(x)=\int_{a}^{b}x(t)y_{0}(t)dt $$ for all $x = x(t) \in C[a,b]$, ...
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9 views

about analytic operator valued function

What's the definition of an analytic operator valued function ? Can we consider an analytic operator valued function as analytic function?