Articles of functional analysis

Spectral Measures: Spectral Spaces (I)

Problem Given a Hilbert space $\mathcal{H}$. Let the Lebesgue measure be $\lambda$. Consider a Borel spectral measure: $$E:\mathcal{B}(\mathbb{C})\to\mathcal{B}(\mathcal{H})$$ Denote its probability measures by: $$\nu_\varphi(A):=\|E(A)\varphi\|^2$$ Introduce the spectral space: $$\mathcal{H}_\parallel:=\{\varphi:\nu_\varphi\ll\lambda\}$$ $$\mathcal{H}_\perp:=\{\varphi:\nu_\varphi\perp\lambda\}$$ Then they decompose: $$\mathcal{H}=\mathcal{H}_\parallel\oplus\mathcal{H}_\perp$$ How to prove this? Attention This thread has been split: Spectral Spaces (II)

$a\mapsto \log\left(\lVert f\lVert_{1/a}\right)$ is a convex map

Proof for $a\in (0,1)$ that $a\to F(a)= \log\left(\lVert f\lVert_{1/a}\right)$ is a convex map, if $f\in L^p(X)$ for all $p\geq1$ in some measure space $(X,\mu)$. I’d like to prove that for all $a_1,a_2\in(0,1)$ and $t\in[0,1]$ the following holds: \begin{align} F(t a_1+(1-t)a_2)\leq t F(a_1)+(1-t)F(a_2)\end{align} using the properties of the logarithm yields $F(a)=a \log\left(\int_X \lvert f\lvert^{1/a}\right)$. Plugging this […]

If $f \in L^2$, then $f \in L^1$ and$\|f\|_{L^1} \leq \sqrt{2\pi} \|f\|_{L^2}$

I’m learning about Fourier analysis and need help with the following problem (which is part of a subchapter on $L^p$ spaces): Using the Cauchy-Schwarz inequality show that if $f \in L^2[-\pi, \pi]$, then $(1)$ $f \in L^1[-\pi, \pi]$ and $(2)$ $\|f\|_{L^1} \leq \sqrt{2\pi} \|f\|_{L^2}$. My work and thoughts: $(1)$ We note that if $f \in […]

Character space of $\mathcal P(K)$

Let $K$ be a compact subset of $\Bbb C$. Let $\mathcal P(K):=$ closure in $||\cdot||_{\infty}$ of all complex polynomials on $K$. Note that $\mathcal P(K) = C(K)$ do not generally hold since Stone-Weierstrass do not apply to complex polynomial. What is the character space $\Phi_{\mathcal P(K)}$ of $\mathcal P(K)$? It is commonly known that $\Phi_{C(K)}=\{ […]

Bochner Integral: Approximability

Problem Given a measure space $\Omega$ and a Banach space $E$. Consider a Bochner measurable function $S_n\to F$. Then it admits an approximation from nearly below: $$\|S_n(\omega)\|\leq \vartheta\|F(\omega)\|:\quad S_n\to F\quad(\vartheta>1)$$ (This is sufficient for most cases regarding proofs.) Can it happen that it does not admit an approximation from below: $$\|S_n(\omega)\|\leq\|F(\omega)\|:\quad S_n\to F$$ (I’m just […]

Continuity of bounded and convex function on Hilbert space

I’m looking for the proof (or at least hints how to prove it) of theorem: Let $H$ be a real Hilbert space and let $f:H\to (-\infty,+\infty]$ be a convex function, bounded from above in some neighbourhood of $x\in H$. Then $f$ is continuous in that point. Thanks in advance, K.

Linear Functional: Continuous?

This question already has an answer here: At most finitely many (Hamel) coordinate functionals are continuous – different proof 2 answers

A completely continuous operator carries weakly Cauchy sequences into norm-convergent ones

Let $X,Y$ be Banach spaces. Show that if $T:X\to Y$ is a completely continuous operator, then $T$ carries weakly Cauchy sequences into norm-convergent sequences. Let $(x_n)_{n=1}^\infty$ be a weakly Cauchy sequence. Let us suppose that $(Tx_n)_{n=1}^\infty$ is not norm-convergent, i.e., it is not a Cauchy sequence. Then I don’t know what can I do. A […]

Semigroups: Entire Elements (I)

Problem Given a Banach space $E$. Consider a C0-semigroup: $$T:\mathbb{R}_+\to\mathcal{B}(E)$$ Define its generator by: $$Ax:=\lim_{h\downarrow0}\frac{1}{h}(T(h)x-x)\in E$$ (It is a densely-defined closed operator.) Denote the convergence radius by: $$\rho_x:=\left(\limsup_{k\to\infty}\sqrt[k]{\frac{1}{k!}\|A^kx\|}\right)^{-1}$$ Generate a semigroup via Taylor series: $$\rho_x=\infty:\quad e^{tA}x:=\sum_{k=0}^\infty\frac{1}{k!}t^kA^kx$$ Can it happen that it has no entire vectors at all? Reference This is the start-up for: Semigroups: Entire […]

Polar Decomposition: Unitarity

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).