# Three basic questions about linear operator in a Hilbert space

Just come across three questions in reading a paper.

Suppose we are dealing with a Hilbert space of $L_{2}[0,1]$ and all the functions mentioned below are in $L_{2}[0,1]$.

Define the operator $A$ by $$(Av)(w):=\int^{1}_{0}v(x)f_{XW}(x,w)dx.$$ Let $\left\{ \psi_{j}: j=1,2,... \right\}$ be a complete, orthonormal basis for $L_{2}[0,1]$, so that $$f_{XW}(x,w)=\sum_{j=1}^{\infty}\sum_{k=1}^{\infty}c_{jk}\psi_{j}(x)\psi_{k}(w).$$ The paper says: Let $A_{n}$ be the operator on $L_{2}[0,1]$ whose kernel is $$a_{n}(x,w)=\sum_{j=1}^{J_{n}}\sum_{k=1}^{J_{n}}c_{jk}\psi_{j}(x)\psi_{k}(w).$$ My first question is: Does this mean that $A_{n}$ is an operator such that $$(A_{n}v)(w):=\int^{1}_{0}v(x)a_{n}(x,w)dx?$$

It also says $$(A_{n}^{-1}\psi_{k})(x)=\sum^{J_{n}}_{j=1}c^{jk}\psi_{j}(x),$$ where $c^{jk}$ is the $(j,k)$ element of the inverse of the $J_{n}\times J_{n}$ matrix $[c_{jk}]$.

My second question is that: Is this because $A_{n}^{-1}$ has kernel as $$\sum_{j=1}^{J_{n}}\sum_{k=1}^{J_{n}}c^{jk}\psi_{j}(x)\psi_{k}(w)?$$

My third question: is this true that $$\sup_{\parallel h \parallel \neq 0} \frac{\parallel h \parallel}{\parallel(A^{\ast}A)^{1/2}h\parallel} =\sup_{\parallel h \parallel \neq 0} \frac{\parallel h \parallel}{\parallel Ah\parallel}?$$

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With regards to your third question, $$\|(A^\ast A)^{1/2}h \| = \langle (A^\ast A)^{1/2}h, (A^\ast A)^{1/2} \rangle^{1/2} = \langle h, A^\ast A h \rangle^{1/2} = ?$$ –  Branimir Ćaćić Dec 14 '13 at 7:52
I see, Braminir. Thank you. –  Jie Wei Dec 14 '13 at 9:19