# fa.functional-analysis's questions - English 1answer

5.329 fa.functional-analysis questions.

### 21 Symmetric strengthening of the Cauchy-Schwarz inequality

In this great question by Nathaniel Johnston, and in its answers, we can learn the following remarkable inequality: For all $v,w \in \mathbb{R}^n$ we have \begin{align*} \|v^2\| \, \|w^2\| - \langle ...

### 3 Dixmier traces, Wodzicki residue and residues of zeta functions

Let $M$ be an $n$ dimensional closed manifold and consider an elliptic, pseudodifferential operator $P$ of order $-n$. Here are some facts which I had learned so far: 1. There exists a density defined ...

### 2 If you can approximate vectors well, can you do so parametrically?

Let $B$ be a Banach space, and $V$ a topological vector space with topology induced by some translation-invariant metric $d$, equipped with a continuous linear map $p: B \to V$ with dense image. ...

### 4 compact embedding for Sobolev spaces

The Sobolev embedding $H_{0, rad}^{1}(\Omega) \hookrightarrow L^{p+1}(\Omega)$ is compact for all $p>1$ where $\Omega= \{x\in \mathbb R^N: 0<a<|x|<b\}.$ Is it possible to determine the ...

### 31 Can $L^p(\mathbb{R})$ and $L^q(\mathbb{R})$ be isomorphic?

1 answers, 2.598 views fa.functional-analysis banach-spaces
Let $p,q \in (1,\infty)$ with $p\neq q$. Are the Banach spaces $L^p(\mathbb{R})$, $L^q(\mathbb{R})$ isomorphic?

### 9 Does the Banach space $( \ell ^2 \oplus \ell ^2 )$ have F.P.P?

The space $( \ell^2 ,\lVert \cdot \rVert _2 )$ is a Hilbert space. The space $X=(\ell^2 \oplus \ell^2 , \lVert \cdot \rVert_\infty )$ is a Banach space. Does X have fixed point property? (For any ...

### 3 eta invariant and spectral flow

We know that for a family of first-order self adjoint elliptic (Fredholm) operator $A_t$, for $t\in [0,1]$ we have the formula $$\eta(A_1)-\eta(A_0)=spfl(A_t)_{t\in[0,1]}+\int^1_0 \omega(s)ds,$$ where ...

### 8 Maps which are both completely positive and positive

Definition:A linear map $f:\mathbb C^n\to \mathbb C^n$ is called positive if $\langle fa,a\rangle\ge0$ for all $a\in \mathbb C^n$. Equivalently, $f\in M_{n}(\mathbb C)$ is positive if it can be ...

### Supremum over all invariant Borel probability measures of the ergodic averages ratio of rates

Let $M$ a two-dimensional compact manifold and $f:M\to M$ a diffeomorphism $C^r$, $r\geq 2$ and $f(x,y)=(mx,\lambda y)$ where $m:M\to \mathbb{R}$ and $\lambda:M\to \mathbb{R}$ ,$\lambda<1<m$. ...

### 4 Power series in functions other than monomials

I would like to understand how approximations by monomials and approximations by other kinds of functions are related which I illustrate with an example. Consider the interval $[-\pi,\pi]$ let's say. ...

### $\sup_{f} \inf_{z\in D} [f_x^2(z)+f_y^2(z)]$ for $|f|\leq1$ on a unit disk

Let $f:\mathbb{R^2}\mapsto\mathbb{R}$ be continuous and have partial derivatives in $D=\{(x,y):x^2+y^2\leq1\}$, and let $\mathscr{H}$ the set of such functions for which $\sup_D |f|\leq1$. Could ...

### 2 Compact restrictions of the inclusion of $J:L_\infty(0,1)\to L_1(0,1)$

Given Banach spaces $X$, $Y$ and a bounded operator $T:X\to Y$ with non-closed range, a perturbation argument shows that there exists an infinite-dimensional closed subspace $M$ of $X$ such that the ...

### Convergence of self-adjoint elements in $\sigma$-weak topology

In a von Neumann algebra, if $A_{\alpha}$ converges to $0$ in the $\sigma$-weak topology, do the positive parts $(A_{\alpha})_{+}$ necessarily converge to $0$ in the $\sigma$-weak topology?

### 1 Compact Embedding Between Parabolic Holder Spaces

My question is about the following compact embedding: $$C^{\sigma+2, \sigma/2+1}_{x, t}(Q_T)\hookrightarrow C^{\sigma, \sigma/2}_{x, t}(Q_T).$$ what condition should be put ...