let $ H$ be a Hilbert space and $ T \in B ( H ) $‎.

Question

Please suggest me a good book on compact normal operators . I have a lot of questions on this subject.I'm Slightly information in this topic.

let $ H$ be a Hilbert space and $ T \in B ( H ) $‎. ‎

a‎: ‎$ T $ is normal if only if $ \forall x ‎\in H ‎\qquad‎ ‎\parallel Tx ‎\parallel = ‎\parallel T‎^{*} x ‎‎\parallel‎‎‎$‎.‎‎ ‎

b:‎ ‎$ T $ is normal ,‎ ‎then‎ ‎$ ‎kern T = ‎‎‎ker‎n ‎T‎^{*} =‎ (‎ ran T‎ )‎ ‎‎^{‎\perp‎}‎‎ $‎.‎

c:‎ ‎$ T \in B( H ) $ ‎is ‎invertible‎, then ‎$ ‎\parallel T ‎\parallel =‎ ‎‎\parallel ‎T‎^{-1}‎‎ ‎\parallel‎ $ ‎if ‎only ‎if‎ ‎$ T $ ‎is a‎ Unitary ‎operator.‎

Answer 1

There are many good books on the subject

• Introductory Functional Analysis with Applications, Erwin Kreyszig. Well-written and good applications.

• Functional Analysis, George Bachman and Lawrence Narici. Great introductory text, and appropriate for self-study, without much background.

• Functional Analysis, Peter D. Lax. Thorough treatment of a large set of standard topics.

For a complex space, the polarization identity $$ (x,y) = \frac{1}{4}\sum_{n=0}^{3}i^n(x+i^ny,x+i^ny) $$ gives a lot of interesting results for complex inner product spaces. For example, if you define $T$ to be normal iff $T^*T=TT^*$, then $T$ is normal iff $$ (T^*Tx,y)=(TT^*x,y),\;\;\; x,y\in H. $$ By polarization this is equivalent to $$ (T^*Tx,x) = (TT^*x,x),\;\;\; x\in H, \\ (Tx,Tx) = (T^*x,T^*x), \;\;\; x\in H, \\ \|Tx\|^2=\|T^*x\|^2,\;\;\; x\in H. $$ By the last statement, $\mbox{ker}(T)=\mbox{ker}(T^*)$ if $T$ is normal because $Tx=0$ iff $T^*x=0$ for normal $T$.