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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542 views

$\operatorname{Range}T$ is a closed subspace.

Let $X,Y$ two Banach spaces. If $T \in \mathcal{B}(X,Y)$ study if $\operatorname{Range}T$ is a closed subspaces. How can I prove this fact ? What theorems can I use ? thanks :)
1
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1answer
56 views

Operator question. $\sigma(T)\neq \varnothing.$

Let $X$ be a Banach real space and $T \in\mathcal{B}(X)$, where $T$ is an operator. Study if: $$\sigma(T)\neq \varnothing.$$ Can you help me please, thanks :)
2
votes
2answers
111 views

Compactness of multiplication operator on $C[0,1]$

Find a condition on function $a\in C[0,1]$ such that the operator $A:C[0,1]\rightarrow C[0,1]$ $$(Ax)(t) = a(t)x(t)$$ is compact? We are taking uniform norm on $C[0,1]$.
2
votes
0answers
254 views

Fredholm and Compact Operators

Let $X$ and $Y$ be Banach spaces and $T\in B(X,Y)$ be Fredholm. Then there is $S\in B(Y,X)$ such that $ST=I+K_{1}$ and $TS=I+K_{2}$ where $K_{1},K_{2}$ are compact operators.Proof: Since $T$ is ...
1
vote
3answers
240 views

If $\Delta$ is the Laplace operator, what are $| \Delta |$ and $|\Delta +1|$

Assuming $\Delta : H^2(\mathbb{R}^n) \to L^2(\mathbb{R}^n)$ be the Laplace operator, then: What is the exact definition of $| \Delta |$? What is $|\Delta +1| $ also? This answer to another ...
2
votes
1answer
278 views

Estimate on the norm of a self-adjoint operator

EDIT: thks to Martin's comment I realize the previous version was wrong. Here is the correct version of what I need to show: I am trying to show that if $A$ is a self - adjoint operator in a Hilbert ...
0
votes
1answer
39 views

$‎\sigma(a)=‎\sigma‎(b)‎$‎‎, ‎‎if ‎‎‎$‎a,b$‎ ‎‎are unitarily equivalent

‎Let ‎$‎A$ be a *-algebra and ‎$‎a,b$ are ‎unitaril‎y equivalent ‎in ‎‎$‎A$ ( i.e. there exists a unitary ‎$‎u$ of ‎$‎A$ s.t ‎$b=uau^{*‎}‎$ ‎‎).‎ ‎I ‎want ‎to ‎prove ‎that ...
1
vote
2answers
207 views

Composition of Fredholm Operators

If $ST$ is a Fredholm operator, then show that $T$ is Fredholm if and only if $S$ is Fredholm.
0
votes
1answer
438 views

There are compact operators that are not norm-limits of finite-rank operators

Given an example of a Banach space for which There are compact operators that are not norm-limits of finite-rank operators. Tanks for answer
1
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0answers
186 views

Integral operators with operator valued kernels

This is the definition for integral operators I know: Let $\Omega \subset \mathbb{R}^n$ and $D \subset \mathbb{R}^n$. Let $K : \Omega \times D \to \mathbb{C}$ be measurable. A linear operator $T: ...
0
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0answers
43 views

$\widehat{a}: \Omega(A)‎\rightarrow‎ \mathbb{C}~,~\tau‎ \mapsto \tau(A)‎‎ $

Suppose that $A$ is abelian Banach algebra for which the space $\Omega(A)$ is non-empty. If $a \in A$, we define the function $\widehat{a}‎‎$ by $$\widehat{a}: \Omega(A)‎\rightarrow‎ ...
4
votes
1answer
55 views

A question about operator representation

Let $H$ be a separable Hilbert space and let $A$ be a compact operator acting on $H$. In general we may write $H = E_A\oplus E_A^\perp$. Let us consider the $2\times 2$ operator matrix of $A$ ...
3
votes
1answer
48 views

About $C_{0}$-semigroup and how is proved that $A\int^{t}_{0}{T(\tau)xd\tau}=T(t)-x.$

Let $\{T_{t}\}_{t \geq0}$ be a $C_{0}$-semigroup and we will denote $$D(A)=\{x\in X : \exists \lim_{h\rightarrow 0_{+}} \frac{T(h)x-x}{h}\}$$ and define $$A:D(A) \rightarrow X; ...
1
vote
1answer
34 views

Gelf‎and ‎representation ‎Theorem

In ‎proof ‎of "‎‎Gelf‎and ‎representation ‎Theorem‎" ‎(see 1.3.6 Theorem of Murphy's book )‎, I ‎am ‎understanding ‎that ‎why ‎the ‎map $$ A ‎‎\rightarrow‎ ‎C_{0}(‎\Omega(A)‎)~ , ‎~‎‎a‎‎\rightarrow‎ ...
6
votes
1answer
226 views

Kernel of adjoint operator

This problem is puzzling me, even though it should be really simple. Let $L=-\partial_x^2 + \frac 1 2 x^{-2}$ be an operator defined on $D(L)=C^\infty_c(0,+\infty)\subset L^2(0,+\infty)$. Its adjoint ...
1
vote
1answer
109 views

Self-adjointness of $D=\frac{d^2}{dx^2}-1$ with boundary conditions $u'(0) = 0 = u'(a)$ on $[0,a]$.

Im trying to show that $$D=\frac{d^2}{dx^2}-1$$ is self adjoint on $[0,a]$ subject to $u'(0)=u'(a)=0$. I think I need to use integration by parts but I'm not sure how to do that.
0
votes
1answer
84 views

Norm of normal Operator A

I just found the following equality $||A||=sup_{\lambda\in\sigma(A)} |\lambda|$ My question: What exactly means $\sigma(A)$ and why this is true ? I always thouht the only way to get the ...
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0answers
65 views

If limit of $f(n)$ is zero then the operator is compact

I want to prove the following: Suppose $\mathfrak{H}$ is the Hilbertspace $l^2(\Bbb{N})$ and $T_f$ the multiplication operator on $\mathfrak{H}$, thus $T_f\psi=f\psi$ for ...
2
votes
1answer
74 views

Why are these two operators similar?

Let $X$ be a Hilbert space with ON-basis $\lbrace e_n : ~ n \in \mathbb{N} \rbrace$. Furthermore let $A, ~ \Gamma : X \to X$ be linear operators with $A e_n = \alpha_n e_n$ and $\Gamma e_n = \gamma_n ...
1
vote
1answer
55 views

Relationship between spectrum and Norm of bounded linear maps .

I am reading the following paper : http://www2.icmc.usp.br/~sma/cadernos/toc9.1/292.pdf In the second paragraph the author introduces a new operator $\|x\|_{T,\epsilon}$ , which i don't really ...
0
votes
0answers
127 views

Operators bounded below on a linearly dense subset

A bounded operator $T\colon X\to Y$ is bounded below if there is $\lambda>0$ such that $\|Tx\|\geqslant \lambda \|x\|$ for all $x\in X$. Suppose $D$ is a linearly dense subset of $X$ such that ...
1
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1answer
51 views

Convergence of spectrum along with the convergence of the Operator.

This seems to be very interesting result , ie If operators $\{A_n\}$ in Banach space $B(X)$ and if $A_n \to A$ , $A \in B(X)$in operator norm then $\lambda_n \in \sigma(A_n)$ ie spectrum of $A_n$ ...
3
votes
1answer
337 views

Computing the spectral decomposition for the multiplication operator $f(x) = \frac{1}{1+x^2}$

I am trying to use the spectral theorem for self adjoint operators to decompose the spectrum of the multiplication operator $f(x) = \frac{1}{1+x^2}$ on $L^2(\mathbb{R}).$ This is a problem in Teschl's ...
1
vote
1answer
157 views

‎If ‎‎$‎X$ is an infinite-dimensional Banach space and ‎‎$‎‎u‎\in ‎B(X)‎$ ,then $\bigcap_{v\in K(X)}\sigma(u+v) =\cdots$

‎If ‎‎$‎X$ is an infinite-dimensional Banach space and ‎‎$‎‎u‎\in ‎B(X)‎$,why the following equality is true? $$\bigcap_{v\in K(X)}\sigma(u+v) =\sigma(u) \setminus \{\lambda \in\mathbb{C}\mid u - ...
-1
votes
1answer
101 views

The ‎inclusion relation $\sigma(ab) ‎\subseteq ‎\sigma(a)‎\sigma(b)$ is not true for all Banach algebras

Let ‎‎$‎A$ ‎be a‎ ‎unital ‎abelian‎ ‎Banach ‎algebra. ‎Give ‎me ‎an ‎example ‎that two ‎following ‎inclusion ‎relations ‎is ‎not ‎true ‎for ‎all ‎Banach ‎algebras‎ $$\sigma(a+b) ‎\subseteq ...
4
votes
1answer
101 views

What is the domain of $\vert \Delta\vert^{1/2}$?

Assume $U$ is a bounded open subset of $\mathbb{R}^N$. Furthermore $\Delta: H^2(U) \subset L^2(U) \to L^2(U)$ is the Laplace operator. My question is: What is the domain of $\vert ...
5
votes
0answers
214 views

Question about the Spectral Theorem for Self Adjoint Operators and Eigenvalues

I have been working through Teschl's book "Mathematical Methods in Quantum Mechanics with Applications to Schrodinger Operators" and I am stuck on a problem in Chapter 3. I am trying to prove that if ...
1
vote
1answer
103 views

Picard criterion: Show $\mbox{range}(T)^{\bot}=\overline{\mbox{range}(T)}^{\bot}$

The so-called Picard criterion is: Let X,Y be Hilbertspaces and $T\colon X\to Y$ is a compact operator with singular value decomposition system $\left\{(\sigma_j,u_j,v_j)\right\}$. An element ...
1
vote
1answer
126 views

Spectral radius of an operator .

I would like to know the spectral radius of the operator $T_k$ from $C[0,1] \to C[0,1]$ : $$T_k x (t)= \int_0^1 k(t,s) x(s) ds$$ where $k(x,y)\colon [0,1]^2 \to \mathbb C$ is continuous. And ...
1
vote
1answer
83 views

Picard Condition (searching for an idea)

The so-called Picard-condition is: Let X,Y be Hilbertspaces and $T\colon X\to Y$ is a compact operator with singular value decomposition system $\left\{(\sigma_j,u_j,v_j)\right\}$. An element ...
4
votes
1answer
70 views

Smallness/ Rigidity of $\kappa(\mathcal{H})$ without using minimal projections?

Let $\mathcal{H}$ be a Hilbert space and $\kappa(\mathcal{H})$ the $C^*$-algebra of compact operators on $\mathcal{H}$. By smallness/ rigidity of $\kappa(\mathcal{H})$ I am referring to the following ...
1
vote
0answers
46 views

The hermitian element $h=\sum_{n=1}^\infty \frac{p_{n}}{3^{n}}$ generates $C_{0}(\Omega)$‎

‎Please help me to solve the following problem‎ : Let $\Omega$ be a locally compact Hausdorff space‎, ‎and suppose that the $C^{*}$-algebra $C_{0}(\Omega)$ is generated by a sequence of projections ...
1
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0answers
83 views

Exponential of an operator plus a constant term

I am reading a book on operator and matrix representation. Most of the examples are on Physics and they mention many terms like 'commute', i.e. the order of application of two operators might be ...
3
votes
1answer
324 views

Eigenvalues integral operator - general case

Let $T$ be an integral operator on $L^2([0,1])$, such that: $$ (Tf)(x) = \int_0^1K(x,y)f(y)dy, $$ with $K(x,y): [0,1]^2 \rightarrow \mathbb{R}$ continuous and $K(x,y) = K(y,x)$, $K(x,y)\geq0$ $ ...
1
vote
1answer
40 views

A Linear map $‎u : X ‎\longrightarrow ‎Y‎‎$ ‎ ‎is ‎not ‎bounded ‎below ‎‎iff ‎there ‎is …

Do you help me to: c‎hecking ‎that a‎‎ ‎linear ‎map ‎‎$‎u : X ‎\longrightarrow ‎Y‎‎$ ‎between ‎Banach ‎spaces ‎is ‎not ‎bounded ‎below ‎if ‎and ‎only ‎if ‎there ‎is a‎ ‎sequence ‎of ‎unit ‎vector ...
2
votes
2answers
115 views

‎‎If $A$ contains ‎an ‎idempotent $e‎$ (‎‎$‎e‎\neq ‎‎0,1‎‎$‎) , then $‎\Omega(A)‎$ ‎is ‎disconnected

If $A$‎ ‎be a‎ ‎unital ‎abelian ‎Banach ‎algebra ‎and ‎contains ‎an ‎idempotent $e$‎ ‎(that ‎is ‎‎$‎e=‎e‎^{‎2‎}‎‎$‎) ‎other ‎than $0$‎ ‎and $1$‎ ,‎ ‎then help me to show that ‎‎$‎\Omega(A)‎$ ‎is ...
1
vote
0answers
98 views

Finding the spectral radius and spectrum .

I am solving the following question : If $k:[0,1]^2\to \mathbb C$ is continuous and $T_k : C[0,1] \to C[0,1]$ such that $$(T_kx)(t)=\int_0^t k(t,s)x(s) ds$$ Define $k_n: [0,1]^2\to \mathbb C$ ...
2
votes
0answers
61 views

Show compactness of an evolution operator

Consider the heat equation $$ u_{t}=u_{xx},~~~~~u_0(x)=u(0,x)$$ with $u\colon [0,T]\times\mathbb{R}\to\mathbb{R}, (t,x)\mapsto u(t,x)$ and the evolution operator $E(T)$ with $E(T)u_0=u(T,x)$. 1.) ...
2
votes
1answer
256 views

A generalization of the Cauchy-Schwarz inequality to linear operators

If $A$ is an operator and $A \in \mathcal{B_{+}(X)}$ (the set of the positive operators) then the generalization of the Cauchy-Buniakowsky-Schwarz inequality holds: $$|\langle Ax,y\rangle| \leq ...
2
votes
1answer
90 views

Why is $\langle Ax, Ax \rangle = \langle A^2 x, x\rangle$?

Let $X$ be a Hilbert space and $A\in \mathcal{B}(X)$ be self-adjoint. How can I prove: $$\langle Ax, Ax \rangle = \langle A^2 x, x \rangle$$ I know it is a simple problem, but I don't know how to ...
2
votes
0answers
237 views

Determining the spectral representation of a operator

The spectral representation for a self-adjoint operator $T \in L(H)$ for H a Hilbert space is written as: $$ T = \sum_{\lambda \in \sigma(T)} \lambda \pi_{\lambda}, $$ where $\sigma(T)$ is the ...
1
vote
0answers
29 views

How to show that density?

Show $$ \overline{\operatorname{span}(v_j)}=L^2([0,1]),~~~~~\overline{\operatorname{span}(u_j)}=L^2([0,1]) $$ with $$ v_j(x)=\sqrt{2}\cos((j-1/2)\pi x),~~~~~u_j(x)=\sqrt{2}\sin((j-1/2)\pi x). $$ ...
3
votes
1answer
65 views

Spectrum of operator on canonical orthonormal system

Define the operator $T: l^2 \rightarrow l^2$ on the canonical orthonormal system $(e_k)_k$ by: $$ Te_k := \frac{e_k}{k} + \frac{e_{k+1}}{k+1}, $$ such that for $a\in l^2$: $$ T((a_i)_i) = (a_1, ...
2
votes
0answers
78 views

Verify a given SVD of an operator

Show that the Singular Value Decomposition of the operator $$ A\colon L^2([0,1])\to L^2([0,1]), x\mapsto\int\limits_0^t x(s)\, ds $$ is given by $$ ...
4
votes
2answers
96 views

If $A\leq B$ in the sense of quadratic forms, then must $A^{-1} \geq B^{-1}$?

Let $A$ and $B$ be symmetric invertible operators on a Hilbert space $X$. Suppose $$ \langle Ax , x \rangle \leq \langle Bx , x \rangle $$ for each $x\in X$. Does it follow that $\langle A^{-1} x ,x ...
3
votes
0answers
60 views

A set of trajectories as a linear subspace of Hilbert space

Let $\left\{S(t)\right\}_{0 \leqslant t \leqslant \theta}$ be a strongly continuous semigroup of linear continuous operators in Hilbert space $H$, $S(0) = I$. Let $x$ be some element of $H$. Then its ...
2
votes
1answer
90 views

Evolution operator

We call a function that assigns a starting value of a time-dependent differentialfunction to a solution of a later timevalue as the evolution operator $E(t)$. Look at the thermal equation $$ ...
3
votes
1answer
86 views

Singular Value Decomposition - what do I have to do?

Show that the Singular Value Decomposition of $$ T\colon L^2([0,1])\to H^1([0,1]), x\mapsto\int\limits_0^t x(s)\, ds $$ is given by $$ \sigma_j=\frac{1}{(j-1/2)\pi}, v_j(x)=\sqrt{2}\cos((j-1/2)\pi ...
3
votes
2answers
227 views

Find adjoint operator of an operator T

I would like to find the adjoint operator of $$ T\colon L^2([0,1])\to H^1([0,1]),\quad x\mapsto\int\limits_0^t x(s)\, ds. $$ Here $H^1([0,1])$ is the Sobolev space $W^{1,2}([0,1])$. I tried to find ...
-3
votes
1answer
104 views

If a,b ‎are ‎unitary ‎equivalent,‎Dose ‎ ‎‎$‎\sigma(a)=‎\sigma(b)‎$‎ is true?

‎‎Let A‎ ‎is ‎an ‎unital‎‎ ‎algebra ‎and ‎‎$ ‎Ad‎~u:‎‎‎A\rightarrow ‎A~,~a‎\mapsto~‎uau‎^{*}‎‎$ ‎and u‎ ‎is ‎unitary ‎element ‎of A‎(‎$‎uu‎^{‎*‎}=‎u‎‎^{*}‎u=1‎$‎), ‎if ‎‎$‎b=‎uau‎^{‎*‎}‎‎$ ‎(a,b ‎are ...