1
vote
0answers
12 views

Show that $B(X)$ is semisimple for a Banach space $X$

Show that $B(X)$ is a semisimple Banach algebra, where $X$ is a Banach space. That is, to show that rad $B(X)=\{0\}$, or equivalently, to show $\sigma(AT)={0} \, \forall T\in B(X)\Rightarrow A=0$ I ...
5
votes
1answer
60 views

Question about Hahn-Banach theorem

Let $(X,\|\cdot\|_1)$ and $(Y,\|\cdot\|_2)$ be normed spaces, and $X\subset Y$. If each $f\in (X,\|\cdot\|_1)^\ast$ extends to a bounded linear functional in $(Y,\|\cdot\|_2)^\ast$ with same norm, ...
1
vote
0answers
35 views

Compact and bounded if and only if $X$ is finite dimensional

I tried to prove the theorem Let $X$ be a Banach space. Then $K(X) = B(X)$ if and only if $X$ is finite-dimensional. Please could someone check my proof? Let us use the fact that a linear ...
1
vote
1answer
18 views

Show compactness of an operator with Arzelà–Ascoli

We have $K\colon L^{2}(a,b) \rightarrow L^{2}(a,b)$ such that $ Kf(t)=\sum_{j=1}^{n}\phi_{j}(t) \int_{a}^{b} \psi_{j}(S) f(s)ds$ where $\phi_{j} ,\psi_{j} \in L^{2}(a,b)$. We want to show that K is ...
1
vote
0answers
50 views

Compact operators is a linear subspace of bounded operators

Let $X,Y$ be Banach spaces. Let $B(X,Y)$ be the set of bounded linear operators and let $K(X,Y)$ be the set of compact linear operators. I want to prove that $K(X,Y)$ is a vector subspace of ...
1
vote
1answer
32 views

Invertible iff Bounded below and dense range

Statement: Given a Hilbert space $\mathscr{H}$ and $\mathscr{K}$ and a bounded operator $A \in \mathscr{B}(\mathscr{H}, \mathscr{K})$. Show that $A$ is invertible if and only if $A$ is bounded below ...
1
vote
1answer
19 views

Check continuity of linear functionals and find norms

1) $c_{00} \owns (x_n) \mapsto \sum_{n=0}^{\infty} x_n \in \mathbb{K}$ where $c_{00}$ is a space of sequences that are eventually equal to $0$ with sup norm 2) $\ell^\infty \owns (x_n) \mapsto ...
-1
votes
1answer
50 views

norm of integral operator in $C([0,1])$

If we define on $C([0,1])$ the operator $$ Tf(x) = \int_{0}^{1} K(t,s) f(s) ds$$ where $K$ is a continous function on two variables. I want to show that: $1)$ $||T|| = \displaystyle\max_{t} ...
2
votes
2answers
51 views

Proof of equivalent characterizations of compact operators

As an exercise I tried to prove the following theorem: If $X,Y$ are Banach spaces and $u \in B(X,Y)$ is a bounded linear operator then the following are equivalent: (1) $u$ is compact ...
1
vote
0answers
18 views

Limit of exp of self-adjoint operator

Let $A$ be self-adjoint (possibly unbounded) operator on Hilbert space $\mathcal{H}$. Under what conditions $w-\lim_{t\rightarrow\infty} e^{i A t}=P_0$, where $w-\lim$ - the limit in weak operator ...
-1
votes
1answer
36 views

Need help proving the equivalence of two norms !

Hey I could use alot of help with this problem please! Let (X, <-,->) be a Hilbert space over R. Then, let A: X -> X be a linear operator. Suppose that A is symettric and positive definite. Let ...
0
votes
0answers
14 views

Pseudo-monotone operators research paper question

Hi I just want to know if anyone can see how the result (2.34) is obtained in the following research paper http://caa.epfl.ch/publications/9-Boccardo-Dacorogna1984.pdf. Thanks, I know that it is a ...
-1
votes
2answers
29 views

Bijective Bounded Operator Extension: Where do the new elements go to?

Given a dense, proper subset of complete spaces: $$X,Y\text{ both complete and }A\subsetneq\overline{A}=X$$ and an operator between them: $$T:A\to Y\text{ continuous, linear and bijective}$$ Now, ...
1
vote
2answers
52 views

Isometry <=> Adjoint left inverse [duplicate]

Is it true that: $$T\text{ isometric}\iff T^*\text{ left inverse}$$ Obviously: $$\text{"}\Rightarrow\text{": }\langle x,\tilde{x}\rangle=\langle Tx,T\tilde{x}\rangle=\langle x,T^*T\tilde{x}\rangle$$ ...
-1
votes
2answers
38 views

Isometry: Adjoint = Leftinverse

Given an isometric operator is it true that its adjoint is necessarily leftinverse? My attempt goes like this: $$\langle x,\mathbb{1}\tilde{x}\rangle=\langle x,\tilde{x}\rangle=\langle ...
1
vote
0answers
31 views

Norm of a projection in $L_p$

Let $\mu$ be a probability measure on $[0,1]$ and let us define the projection $P$ on $L^p(\mu)$: $$P \; \colon f \mapsto \mathbb{E}(f){\bf 1}$$ (where ${\bf 1}$ is the constant 1 function). What is ...
2
votes
1answer
53 views

is this true for Hilbert space direct sum of $H$ when $H$ is infinite dimensional?

Let $(H_{\alpha})_{{\alpha \in I}}$ be a $I-$indexed family of Hilbert spaces over $\mathbb{F}$. let $H=\bigoplus H_\alpha$ be their Hilbert space direct sum. Can we say $\dim ...
0
votes
1answer
13 views

The index of some perturbation about elliptic operator with Robin boundary condition

Let $I$ be an closed interval $[0, 1]$. $C^{2}(\bar{I})$ is the space of all $C^{2}$ functions on $(0, 1)$ with continuity at boundary and usual maximal norm. $C(\bar{I})$ is the space of all ...
1
vote
0answers
12 views

Showing a particular integral operator is trace class

Let $f$ and $P$ be continuous, integrable functions $\mathbb{R} \to \mathbb{C}$ vanishing at $\pm \infty%$. Concisely, $f,P \in C_0(\mathbb{R}) \cap L^1(\mathbb{R})$. Also, assume that $P$ is ...
1
vote
1answer
20 views

Integral's limit

Let $X$ be a Banach space and $A$ is a linear bounded operator on $X$. It is well known that for $|\lambda|> \|A\|,$ we have $$\|(\lambda I - A)^{-1}\| \leq \frac{1}{|\lambda|-\|A\|}.$$ Now, let ...
2
votes
1answer
31 views

The image of an bounded and linear operator between two banach spaces is closed if the image is finite codimension?

So here is my problem, I would like to prove the following, Let $X,Y$ be Banach-Spaces and $T:X\rightarrow Y$ a linear and bounded operator. Then $TX$ is closed if it is of finite codimension i.e ...
4
votes
0answers
70 views

How to get a grip on codimensions

I am trying to find a proof for the following problem: Let $X,Y$ be Banach spaces $A,B:X \rightarrow Y$ are bounded linear operators and $Ran(A)$ is closed. If $\left \| Bx\right \|<\left \| ...
2
votes
1answer
21 views

Fast argument to see that the dual map of a projection is a projection

If $X$ is a Banach space and $U,V$ are closed subspaces, such that $X \cong U \oplus V$, then a continuous linear map $P:X \rightarrow X$ is called a projection if $P|_U =id$ and ker(P)=V. Now we ...
1
vote
0answers
18 views

Conjugating an operator with a gauge transformation; how is the kernel affected.

For the differential operator $$ D := i I \frac{d}{dx} + A(x) \colon C^\infty_T([0,\beta]),\mathbb{C}^m) \to C^\infty_T ([0,\beta],\mathbb{C}^m) $$ where $A(x)$ is Hermitian and $C^\infty_T ...
0
votes
0answers
95 views

Spectral properties in Hilbert Space

If $X$ is a Hilbert space, then prove that $$σ(T^*)=\overline{σ(T)}\hspace{2cm}\text{and}$$ $$[R_\lambda(T)]^*=[R_\overline{\lambda}(T)]\hspace{1cm}\text{for all $\lambda\in\rho(T)$ ,}$$ where ...
1
vote
1answer
60 views

Show that any compact set in $\mathbb{C}$ is the spectrum of an operator.

I have been looking around for an example of a general continuous (bounded) linear operator, who's spectrum is any compact set $K\subset\mathbb{C}$. I have seen an example, where we take the set ...
0
votes
0answers
19 views

Confusion with proving that some subspace of a Banach-Space is closed

So here is my problem, I am trying to show that, Let $X,Y$ be Banach-Spaces and $T:X\rightarrow Y$ a linear and bounded map. Then $T(X)$ is closed if $Y/T(X)$ is of finite dimension. While ...
2
votes
1answer
32 views

Equivalent conditions for composition to be compact operator

I did some exercises in Conway's functional analysis book and found the following problem: Let $\tau:[0,1]\to [0,1]$ be continuous and define $A:C[0,1]\to C[0,1]$ by $Af:= f\circ \tau$. Give ...
0
votes
0answers
17 views

Perturbation of resolvent

Let $H$ be a Hilbert space.$A$ is densely defined closed operator and $B$ is $A$-bounded:$||Bx||\leq a||Ax||+b||x||$.Let $\lambda\in\rho(A)$,such that ...
0
votes
0answers
15 views

Domain of the quantum free Hamiltonian in 1D

Consider the quantum free Hamiltonian $H_0 =-\frac{d^2}{dx^2}$ (the Laplacian on the real line). I want to show that it is (essentially) self-adjoint in its domain of definition. The usual approach ...
1
vote
2answers
24 views

About locally convex space

Is a Banach space a locally convex space? Why? Recall A locally convex space is a linear topological space in which the topology has a base consisting of convex sets.
0
votes
0answers
24 views

About inverse of an operator

Let $ X_{p}:= L_{p}([-a,a]\times[-1,1], dxdv), (a>0,\, 1\leq p<\infty)$ and the operator $$\left\lbrace \begin{array}{l} S :X_{p} \rightarrow X_{p} \\ \qquad \psi \mapsto ...
0
votes
1answer
42 views

A question about utilizing Hahn-Banach theorem

There is a quotation below: Let $A$ be a Banach space, $\mathbb{B}(A)$ be the space of all bounded linear maps from $A$ to $A$ and $C\subset \mathbb{B}(A)$ be any convex set. If a net ...
0
votes
1answer
39 views

Composing Projections on a Hilbert Space

Let $P,Q$ be projections on a Hilbert space such that $PQ$ is a projection. I have been able to prove that $PQ=QP$. I want to show that $ker(PQ)$ is contained in $ker(P)+ker(Q)$. If there's a ...
0
votes
0answers
46 views

Self adjointness for functionals

I have posted this question already in the physics forum, but actually nobody could help. I am sorry, this question is related to quantum field theory. The Schrödinger equation of a free scalar field ...
-1
votes
1answer
32 views

Composition property of the functional calculus

Prove from the spectral theorem for normal operators $T$ that for bounded borel functions $f,g$ we have $f{\circ}g(T)$=$f(g(T))$.
0
votes
2answers
72 views

Proving this operator is not closed

I was looking for an example of an operator on a Hilbert space which is not closed. This is what I have done so far, but am not sure if the proof is alright. Consider $H=L^{2}[0,1]$ and $T$ defined ...
3
votes
2answers
42 views

Why $ \|x^* x \| = \|x\|\|x^*\|$ is equivalent to $\|xx^*\| = \|x\|^2$ in the definition of $C^*$ algebra?

I read the definition of $C^*$ algebra in Wikipedia where it says $\|x^* x \| = \|x\|\|x^*\|$ is equivalent to $\|xx^*\| = \|x\|^2$ but I do not know why. Can you show me how to derive $\|xx^*\| = ...
1
vote
3answers
51 views

polar decomposition proof

Let $H$ be a hilbert space and $T$ a bounded linear operator on $H$. I'm trying to prove that there is a partial isometry $V$ on the closure of $Im(|T|)$ such that $T=V|T|$ and $|T|=V^*T$, where ...
2
votes
1answer
53 views

show that the operator $T:l^2\rightarrow V$ is bounded

Let $V$ be the Banach space of all sequences $v=\{\eta_j\}_{j=1}^\infty$ such that $\lim_{j\rightarrow\infty}\eta_j$ exists. The norm on $V$ is given by $\|v\|=\sup_{j\in \mathbb{N}}\eta_j$. Consider ...
1
vote
2answers
58 views

uniqueness of positive operator

Let $A,B$ be commuting positive operators on a hilbert space such that $\langle(A-B)(A+B)x,x\rangle=0$ for all $x$ in the hilbert space. Prove that $A=B$. My attempt: The above implies that $A=B$ on ...
2
votes
1answer
26 views

Regularity estimates for parametrized family of elliptic operators

Let $\Omega\subset\mathbb{R}^3$ be a bounded Lipschitz domain. Suppose we have a parametrized family of linear operators $\{A_\varepsilon\ :\ \varepsilon\in\mathbb{R}_{\geq0}\}$ such that ...
3
votes
1answer
44 views

An exercise of positive element in C*-algebra

Let $A$ be a unital C*-algebra and $\{b_{n}\}$ be a positive invertible sequence in $A$. If $||1_{A}-b_{n}||\rightarrow 0$, can we conclude $||1_{A}-b_{n}^{-\frac{1}{2}}||\rightarrow 0$ ?
1
vote
0answers
27 views

Lack of a polar decomposition

Prove that the left and right shifts on $l_{2}$ have no polar decomposition (i.e. $UP$ where $U$ is unitary and $P$ is positive).
2
votes
1answer
58 views

square root of positive operators

It $T, S$ are positive operators, do we have that $\sqrt{TS}=\sqrt{T}\sqrt{S}$? Are there any basic rules that hold for square roots of positive numbers that don't hold for positive operators?
3
votes
0answers
71 views

A problem of weak* continuity in relation with semigroups

Let $(\Omega,\Sigma,\mu)$ be a probability space. Let $\mathcal{A}$ ba a $\sigma$-subalgebra of $\Sigma$. We denote by $\mathbb{E} \colon L^\infty(\Sigma) \to L^\infty(\mathcal{A})$ the associated ...
1
vote
1answer
32 views

The spectrum of the operators

Let $X, Y$ be the Banach space, and $T_{1}: X\rightarrow X$ and $T_{2}: Y\rightarrow Y$ be the bounded linear operators. Then what is the relationship between $\sigma(T_{1})$, $\sigma(T_{2})$ and ...
0
votes
2answers
62 views

functional calculus and spectral measure

Let $T$ be a normal operator and $f$ be a bounded borel function on ${\sigma}(T)$. If $E_{T}$ and $E_{f(T)}$ are the spectral decompositions of $T$ and $f(T)$ respectively, prove that for any borel ...
0
votes
0answers
13 views

Operator's comparison

I have found the following two concepts: $\bullet$ Let $L$ be a linear operator in a Hilbert space $H$. The operator $B$ is said to be $L$-compact if $D(L)\subset D(B)$ and for any ...
0
votes
2answers
54 views

generalized eigenspace direct sum

Similar to the way an infinite dimensional hilbert space can be written as a direct sum of eigenspaces of a normal compact operator, I was wondering whether it can be written as a direct sum of ...