6
votes
0answers
196 views

Closest fixed point to a convex set

Consider the compact convex sets $Y \subset X \subset \mathbb{R}^n$, and a Lipschitz continuous function $f : X \rightarrow X$. Assume that $f$ has multiple fixed points. (From Brouwer's theorem, $f$ ...
0
votes
2answers
41 views

How find two differential operator $A$ and $B$ such $A\circ \dfrac{d}{dx}=B\circ x$

Question: show that there exists differential operators $A$ and $B$ where $$A=\sum_{k=0}^{n}a_{k}(x)\dfrac{d^k}{dx^k}\neq 0,B=\sum_{k=0}^{n}b_{k}(x)\dfrac{d^k}{dx^k}\neq 0$$ ...
1
vote
1answer
30 views

self-adjoint operator without eigenvalues?

I have a self-adjoint operator $d$ which acts on vector fields defined on $\mathbb{R}^n$. I am interested on its eigenvalues. That is, I study the equation $d(X)-\lambda X=0$. I have found that if ...
3
votes
1answer
47 views

If $T$ is self-adjoint, is the set of power series in $T$ closed?

If $T$ is a bounded self-adjoint operator on a Hilbert space, is the set of convergent power series in $T$ closed in the norm topology? I ask because I'm reading some spectral theorems and I was ...
1
vote
1answer
34 views

Lipschitz constants of projections

Consider two compact sets $A, B \subset \mathbb{R}^n$. Assume that the projection mappings $P_A: \mathbb{R}^n \rightarrow A$, $P_B : \mathbb{R}^n \rightarrow B$ have Lipschitz constant $1$ and $L$, ...
0
votes
0answers
24 views

Fixed point of projected operator

Let $X \subset \mathbb{R}^n$ be a compact convex set and let $f: X \rightarrow X$ be Lipschitz continuous and such that $$ \left( f(x) - f(y) \right)^\top \left( x-y\right) \leq 0 $$ for all $x, y ...
0
votes
0answers
13 views

Coordinate Change Operator

Let $ f: \mathbb{R} \rightarrow \mathbb{R} $ be analytic. Recall that for $ h \in \mathbb{R} $, the translated function $ \tilde{f} (x) = f(x+h) $ can be formally written as $ \tilde{f} = e^{ h ...
2
votes
1answer
46 views

Definition of exponential for operators

if I have a self-adjoint operator $T:D(T) \rightarrow L^2$, then I define its unitary exponential operator by $$e^{iT}(f) := \lim_{k \rightarrow \infty} e^{iT_{k}}(f),$$ where $T_k(f):=\frac{1}{2} ...
3
votes
1answer
57 views

Is $AA^*$ and $A^*A$ self-adjoint?

if I have a densely defined closed linear operator $A$ and $A^* = -A$(same domain also closed). Is this sufficient that $AA^*$ and $A^*A$ are proper self-adjoint operators, assuming that we can also ...
4
votes
2answers
107 views

Spectral theory - continuous spectrum

imagine that I have some differential operator $D$ that is defined on an interval $[a,b]$. Now, assume that we take the boundary conditions in such a way that this operator is self-adjoint. Then, I ...
1
vote
1answer
43 views

Fredholm index for 1-d Schroedinger operator

if I look at a Schroedinger-operator $-\frac{d^2}{dx^2} +V$ on a compact intervall $[a,b] \subset \mathbb{R}$ and I take boundary conditions that this operator is self-adjoint (for example periodic ...
1
vote
1answer
101 views

Spectral theory

I have absolutely no idea about Spectral theory and want to ask some fundamental questions. 1.) What does it mean that the resolvent of an operator is Hilbert-Schmidt? (Cause I saw a theorem that was ...
1
vote
1answer
35 views

unbounded self-adjoint operator

Given an operator $T:D_1(T)\subset L^2 \rightarrow L^2$ and the same operator $T:D_2(T) \subset L^2 \rightarrow L^2$, such that the operator is both times self-adjoint and closed, with $D_1(T) \subset ...
0
votes
0answers
20 views

Operator differentiability

I was wondering, what techniques can one use to prove that an operator (let's say acting on real analytic functions and taking values in a Banach space) is infinitely differentiable? I know that, for ...
1
vote
1answer
31 views

Functional defined on the space of functions with compact support

Let $X=C_c(\mathbb{R})$, the space of functions with compact support, normed with the max norm. Define $\Gamma: X \rightarrow \mathbb{R}$ as: $$ \Gamma f= \int_{- \infty}^{\infty} f(t)dt ...
2
votes
0answers
26 views

The adjoint of unbounded operators as a function.

Let $H_1$, $H_2$ be two possibly distinct real or complex Hilbert spaces, with linearity in the first coordinate of the inner product for concreteness. Let's think of passage to the adjoint as a map ...
0
votes
1answer
36 views

Exercise on isometry

Let $X$ be a Banach space and $T$ a linear bounded operator defined on $L(X,Y)$ with $Y$ a normed space. If $T$ is an isometry then $TX$ is a closed subspace of $Y$. I considered a sequence $y_n$ ...
4
votes
1answer
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 ...
3
votes
1answer
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 ...
0
votes
1answer
19 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 ...
1
vote
1answer
109 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. ...
1
vote
1answer
48 views

Closed Graph Theorem

Let $(x_n)$ be a Schauder basis of $X$ and $(y_n)$ an equivalent one to $Y$. They are supposed to be equivalent, hence for every sequence $(a_n)$ the series $\sum_{n \in \mathbb{N}} a_nx_n$ converges ...
0
votes
0answers
24 views

Saturated Monotone and Increasing Mappings

Let $A : \mathbb{R}^n \rightarrow \mathbb{R}^n$ be a monotone mapping, i.e., $$ \left( A(x) - A(y) \right)^\top \left( x-y\right) \geq 0 $$ for all $x,y \in \mathbb{R}^n$. Let $B : \mathbb{R}^n ...
6
votes
1answer
94 views

Show that $T$ is continuous

I have a question about how to response to: Given a Banach space X and $T: X \rightarrow X^{*}$ a linear operator such that $\langle Tx,x\rangle \geq 0$ for all $x \in X$. Show that $T$ is ...
1
vote
1answer
43 views

Multiplication operator has no no eigenvalues

Statement: Let $M_x$ denote the multiplicative operator acting on $L^2([0,1], \, dx)$ by $M_x(f) = xf$. Show that $M_x$ has no eigenvalues Attempt: Let $M_x(f) = xf$ then we should have $M_x(f) = ...
4
votes
1answer
31 views

Compact operator on invariant subspace is compact

Statement: Let $T \in \mathscr{B}(\mathscr{H})$, where $T$ is a compact operator. Let $M$ be a closed invariant subspace of $T$. Show that the restriction of $T$ to $M$ is compact. Attempted Proof: ...
1
vote
1answer
27 views

Is the composition of monotone operators monotone?

Let $H$ be a real Hilbert space with inner product $\langle\cdot, \cdot \rangle: H \times H \rightarrow \mathbb{R}$, and induced norm $\left\| \cdot \right\|: H \rightarrow \mathbb{R}_{\geq 0}$. Let ...
1
vote
1answer
95 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
22 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
28 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 ...
2
votes
1answer
93 views

Showing that the space of Hilbert-Schmidt operators form a banach space.

How do i show that the set of Hilbert-Schmidt operators $HS(H) = \{T \in B(H) \; : \; \sum^{\infty}_{n=1}\|Te_n\|^2 < \infty \}$ for some countable ONB $\{e_n\}$, on a separable Hilbert Space H, ...
1
vote
1answer
277 views

Spectral Mapping Theorem

Spectral mapping theorem is as follows: https://math.uc.edu/~halpern/Matrix.methods/Homatrixmethods/Spectralmappingthm.pdf Is Spectral mapping theorem true for point spectrum ?
0
votes
1answer
45 views

Closed graph theorem question?

Let $H$ be a Hilbert space. Let $A:\operatorname{dom}A\to H$ has a closed graph, where $\operatorname{dom}A$ is dense in $H$. Let $S\subseteq \operatorname{dom}A$ be dense. Is it true $A_{|S}$ has a ...
7
votes
1answer
158 views

True/False: Self-adjoint compact operator

Let $H$ be a hilbert space and $T$ a compact self-adjoint operator on it. T is also injective on a dense subspace $U \subset H$ and we also have that $T(H) \subset U$. Now I am asked whether it is ...
0
votes
0answers
21 views

Is that operator positive-definite?

Let's consider the integral operator $\phi(x) = \int\limits^1_0\psi(y)\ln\Bigl(\Bigl|\frac{\sqrt{1-x^2}+\sqrt{1-y^2}}{\sqrt{1-x^2}-\sqrt{1-y^2}}\Bigr|\Bigr)\,dy$. How to check is this operator ...
0
votes
1answer
98 views

Proof Riesz representation theorem

I have a question regarding the proof of the Riesz representation theorem. Why do we declare the isomorphism $\Phi: H \rightarrow H'$ in an antilinear way? I mean if, this isomorphism would pick the ...
0
votes
1answer
60 views

Question about step in proof of Schauder's theorem

The statement is the following: Let $X,Y$ be Banach spaces and $T:X \rightarrow Y$ be a continuous linear operator. Then is $T'$ compact iff $T$ is compact. I have already understood the implication ...
2
votes
0answers
86 views

Troublesome proof in Functional Analysis with dual vector space

Greetings to all of you I have tried to prove the following theorem but I am having some troubles with it. Let $X$ be a separable normed space and $(x_n')$ a bounded sequence in $X'$, then there is a ...
3
votes
2answers
72 views

Conjugate convex functions property

$E$ is normed vector space.Let $f\in E^*$ in a bounded linear functional from $E$ to $C$ and fix $x\in E$. We have $$\forall y\in E;\ \ \ \ f(y-x)\leq \frac{1}{2}\|y\|^2-\frac{1}{2}\|x\|^2$$ And I ...
2
votes
1answer
129 views

Relationship between adjoint operators, trace-class operators, compact operators and density operators in Quantum-Mechanics

I don't know much about Functional Analysis, but I was wondering about the following: In Banach spaces it is possible to define for every continuous opertor $T:X \rightarrow Y$ an adjoint Operator ...
0
votes
0answers
29 views

Show, that $T\colon C([a,b])\to C([a,b])$

I have a question concerning an integral equation that is written as an fixed point equation, namely $$ u(x)+\int_a^x F(x,y,u(y))\, dy=f(x,u(x)),~~x\in [a,b] $$ with $$ ...
0
votes
1answer
55 views

Minkowski functional and strange theorem

I have a theorem that says the following: Let X be a normed space and $U\subset X$ a convx subset with $0 \in \text{int(U)}$, then we have: $U$ is absorbing and if $\{x;||x|| < \epsilon\} \subset ...
2
votes
2answers
48 views

Compact operator and limit

I was wondering about something related to compact operators. If we have a compact operator $T:X \mapsto Y$ and a bounded sequence $(x_n)n$, then we know that there is a convergent subsequence ...
2
votes
1answer
66 views

Are these operators and the fourier transform compact?

I do not want a proof but rather an explanation. I just read that $T_k:L^2(\mathbb{R}) \rightarrow L^2(\mathbb{R})$ such that $(T_kf)(s) = \int_{\mathbb{R}} k(s,t)f(t) dt $ is compact. (in this ...
1
vote
1answer
121 views

An example of a non-closable operator

I've encountered the following: Consider the usual Hilbert space $L^2([0,1],dx)$ and the dense subspace $\mathcal{D}=\mathcal{C}[0,1]$. Define $T$ on $\mathcal{D}$ by $T(f)=f(0)$. This is a ...
0
votes
0answers
38 views

Are these linear operators continuous?

For every polynomial $p(t)= \sum_{k=0}^{n} a_k t^k$ we declare its norm by $||a_k||=\sum_{k=0}^{n}|a_k|$. Now, I am supposed to check whether these maps are continuous and in case that they are I ...
4
votes
1answer
74 views

A has an self-adjoint extension

Let $A$ be a symmetric operator satisfying $\langle \phi,A\phi\rangle\geq C\lVert \phi\rVert^{2}$ for all $\phi\in \mathcal{D}(A)$ and some $C\in \mathbb{R}$. Show that the deficiency indiecs are ...
5
votes
0answers
149 views

Dyadic approach to Marcinkiewicz interpolation for Lorentz Spaces

In Exercise 21 , in a note for professor Terrence Tao on his own blog http://terrytao.wordpress.com/2009/03/30/245c-notes-1-interpolation-of-lp-spaces/ Exercise 21 Suppose we are in the ...
9
votes
1answer
75 views

How to prove $e^{\log(1+x)}= 1+x$ by series expansion?

as the title says, i want to prove $e^{\log(1+x)}= 1+x$, by substitute $\log(1+x) = \sum _{i=1} ^{\infty} \frac{(-1)^{i+1}x^i}{i}$ and $e^x=\sum _{i=0} ^{\infty} \frac{x^i}{i!} $. Can some one help ...
1
vote
1answer
90 views

Spectrum of operator in infinite dimensional hilbert space

We know that if a complex hilbert space $H$ is separable, then for every compact set $K$, there exists a bounded linear operator $T : H \to H$ s.t $\sigma (T) = K$. My question is if this still holds ...