Articles of hilbert spaces

Generalized polar decomposition

Let $x\in B(H)$. We say $(x,v,y)$ is a polar decomposition for $x$ if, $\bullet$ $y$ is positive. $\bullet$ $v$ is a partial isometry with $x=vy$. $\bullet$ Ker$(x)$=Ker$(y)$=Ker($v$) The polar decomposition $x=u|x|$ induces a polar decomposition for $x$. These natural problems give rise: Does there exists any (non-trivial) polar decomposition for $x$? Any characterization for all […]

$\bigcup_{n}V_n$ is dense in $V$ implies $\bigcup_{n}L^2(0,T;V_n)$ is dense in $L^2(0,T;V)$?

Let $V$ be a separable Hilbert space with basis $w_j$ and let $V_n$ denote the linear span of $w_j$ for $j=1,…,n$. Clearly $V_n$ are Hilbert spaces and $V_n \subset V_{n+1}$ for all $n$. We have that $\bigcup_{n}V_n$ is dense in $V$. Does it follow that $\bigcup_{n}L^2(0,T;V_n)$ is dense in $L^2(0,T;V)$? I can’t prove it; I […]

Making a complex inner product symmetric

Let $(V, (\cdot, \cdot))$ be a complex inner product space, say a space of complex-valued functions, with $(\cdot, \cdot)$ linear in the second position and sesquilinear in the first. Assume that $V$ is closed under conjugation. I want to prove/disprove that the function $f\colon V\times V\to \mathbb{C}$ defined by $f(x, y)=(\bar{x}, y)$ is symmetric. If […]

Distance between Unilateral shift and invertible operators.

I want to prove that the distance between unilateral shift and normal operators is $1$. But I need to prove that $d(S,\operatorname{Inv}(L(H))= 1$, where $H$ is a Hilbert space. Does anyone have any ideas or hints? T.Y

Hilbert Schmidt operators as an ideal in operators.

Let $H$ be a Hilbert space. For $\{e_n\}$ an orthonormal basis of $H$, we call $T\in B(H)$, a Hilbert Schmidt operator if $ \|T\|_2^2:=\sum_n \|Te_n\|^2 <\infty.$ I have seen somewhere before that $ \|ST\|_2 \leq \|S\|_{B(H)} \|T\|_2 $ for all $S\in B(H)$ arbitrary. In other words, the space of Hilbert Schmidt is an ideal in […]

Question about bases in Hilbert spaces and subspaces

Let $H$ be a Hilbert space. Suppose I have a basis for $H$ called $\{h_j\}$. Define $$H_n := \text{span}\{h_1,…,h_n\}.$$ Suppose now I am given an orthonormal basis for $H$ called $\{v_j\}$. My question is, is it possible to reorder the basis $v_j$ so that 1) For all $i=1,…,n$, the element $v_i \in H_n$ 2) The […]

Two (equivalent ?) norms on Hilbert space

Let $H$ be a vector space equipped with two inner products $\langle \cdot,\cdot\rangle_1, \ \langle \cdot,\cdot\rangle_2$, $(H,\langle \cdot,\cdot\rangle_j)$ is a Hilbert space for $j=1,2$. Suppose that $R_1$ and $R_2$ are linear operators on $H$, which are bounded w.r.t. $\|\cdot\|_1$ and $\|\cdot\|_2$,respectively. Moreover, let us assume that $R_1$ is injective and for any $x,y\in H […]

Hilbert sum of $L_2(X_\nu,\mu_\nu)$ spaces.

Let $\{(X_\nu,\mu_\nu):\nu\in\Lambda\}$ be a family of measurable spaces. Is it true that $\bigoplus_2\{L_2(X_\nu,\mu_\nu):\nu\in\Lambda\}$ isometrically isomorphic to $L_2\left(\bigsqcup\{(X_\nu,\mu_\nu):\nu\in\Lambda\}\right)$. It seems to me that desired isometric isomorphism is $$ i:\bigoplus_2\{L_2(X_\nu,\mu_\nu):\nu\in\Lambda\}\to L_2\left(\bigsqcup\{(X_\nu,\mu_\nu):\nu\in\Lambda\}\right):f\mapsto(x_\nu\mapsto f(\nu)(x_\nu))\qquad x_\nu\in X_\nu,\quad\nu\in\Lambda $$ Here is my proof: $$ \Vert i(f)\Vert^2=\int\limits_{\bigsqcup\{(X_\nu,\mu_\nu):\nu\in\Lambda\}}|i(f)(x)|^2 d\left(\sqcup\{\mu_\nu:\nu\in\Lambda\}\right)(x)= $$ $$ \sum\limits_{\nu\in\Lambda}\quad\int\limits_{X_\nu}|i(f)(x_\nu)|^2d\mu_\nu(x_\nu)= \sum\limits_{\nu\in\Lambda}\quad\int\limits_{X_\nu}|f(\nu)(x_\nu)|^2d\mu_\nu(x_\nu)= $$ $$ \sum\limits_{\nu\in\Lambda}\Vert f(\nu)\Vert^2=\Vert f\Vert^2.$$ Hence $i$ is an isometry. […]

Does every closed, densely defined operator in a Banach space have a closed, densely defined extension on a Hilbert space?

Assume that $H_1,H_2$ are separable Hilbert spaces, $B$ is a separable Banach space and $H_1\subset B\subset H_2$. Assume further that the inclusion mappings are continuous and have dense images. Given any closed, densely defined operator $T$ in $B$, is it then true that there exists a closed, densely defined operator $\tilde T$ in $H_2$ which […]

Prove or disprove that the given expression is “always” positive

I have previously asked a question and I tried to solve it by my own and it led to the question below: Prove or disprove that $$\small\int_{\mathbb{R}}l(y)^xf_0(y)\mathrm{d}y\int_{\mathbb{R}}f_0(y)l(y)^x\ln (l(y)^x)\ln (l(y))\mathrm{d}y-\int_{\mathbb{R}}l(y)^xf_0(y)\ln (l(y))\mathrm{d}y\int_{\mathbb{R}}f_0(y)l(y)^x\ln (l(y)^x)\mathrm{d}y$$ is greater than $0$. Given: $\rightarrow f_0$ and $f_1$ are some density functions $\rightarrow l(y)=\frac{f_1(y)}{f_0(y)}$ is an increasing function. $\rightarrow x\in(0,1)$