I was given a paper by Allan L. Shields on Weighted Shift operators and Analytic Function Theory, American Mathematical Society.
On the very first page, the definition of Weighted Shift operator is defined as follows,
- Introduction. A weighted shift operator $T$ on (complex) Hilbert space $H$ is an operator that maps each vector in some orthonormal basis $\{e_n \}$ into a scalar multiple of the next vector $$Te_n= w_n e_{n+1}$$ for all n.
My problem is, what if $H $ is finite dimensional, how would one define the weighted shift operator on $H$.
Secondly, I don't understand the proof of PROPOSITION 1 as stated in the attachment below. I would so much appreciate if someone help me, give hints or recommend a text. Thank You
