Let (b$_{i,j}$) be a bounded matrix on extit{ l}$^{2}$, $Bbb
T={zinBbb C:|z|=1}$, and A be a bounded matrix on L$^{
2}(mathbb{T)}$ satisfying the conditions
1.$langle Az^{2j},z^{2i}
angle =sigma ^{-1}b_{ij}+|alpha
|^{2}sigma ^{-1}langle Az^{j},z^{i}
angle $;
2.$langle Az^{2j},z^{2i-1}
angle =-alpha sigma
^{-1}b_{ij}+alpha sigma ^{-1}langle Az^{j},z^{i}
angle $;
3.$langle Az^{2j-1},z^{2i}
angle =-overline{alpha }sigma
^{-1}b_{ij}+overline{alpha }sigma ^{-1}langle
Az^{j},z^{i}
angle$;
4.$langle Az^{2j-1},z^{2i-1}
angle =|alpha |^{2}sigma
^{-1}b_{ij}+sigma ^{-1}langle Az^{j},z^{i}
angle $
hspace{-0.76cm} for all $i,jin mathbb{Z}$, where $sigma
=1+|alpha |^{2},,alpha in mathbb{C},alpha
eq0$.
The above conditions evidently suggests that there is a "dyadic"
relation in the entries of $A$. Here in the following picture
illustrates how each $ij-$th entry of $A$ generates the 2 by 2 block
in $A$ with entries ${a_{2i 2j}, a_{2i-1 2j}, a_{2i 2j-1},
a_{2i-1 2j-1}}.$ vspace{-0.3cm}
egin{figure}[hp]
egin{center}
includegraphics[scale=0.42]{cubic.pdf}
end{center}
vspace{-0.8cm}caption{The dyadic recurrent form} end{figure}
It has been shown [2] that $displaystyle A=sum_{n=0}^{infty
}S^{n}BS^{ast n}$, where $Sz^i=sigma ^{-1/2}(overline{alpha
}z^{2i}+z^{2i-1})$ and $$B=sumlimits_{i=-infty}^infty
sumlimits_{j=-infty}^infty b_{ij}(u_{i}otimes u_{j}),
u_{i}(z)=sigma ^{-1/2}z^{2i-1}(alpha -z).$$
In this paper, we shall use the above relations to compute $langle
a_{i,j}
angle $ explicitly.
ewline
Key words: shift operator, bounded matrix, dyadic recurrent formula,
slant Toeplitz operator, separable Hilbert space
2.$langle Az^{2j},z^{2i-1}
angle =-alpha sigma
^{-1}b_{ij}+alpha sigma ^{-1}langle Az^{j},z^{i}
angle $
3.$langle Az^{2j-1},z^{2i}
angle =-overline{alpha }sigma
^{-1}b_{ij}+overline{alpha }sigma ^{-1}langle
Az^{j},z^{i}
angle $
4.$langle Az^{2j-1},z^{2i-1}
angle =|alpha |^{2}sigma
^{-1}b_{ij}+sigma ^{-1}langle Az^{j},z^{i}
angle $
for all $i,jin mathbb{Z}$, where $sigma =1+|alpha
|^{2},,alpha in mathbb{C},alpha
eq0$
egin{figure}[hp]
egin{center}
includegraphics[scale=0.42]{cubic.pdf}
end{center}
caption{The dyadic recurrent form} end{figure}
Since it has been
shown [2] that $displaystyle A=sum_{n=0}^{infty }S^{n}BS^{ast
n}$, where
$ Sz^i=sigma ^{-1/2}(overline{alpha }z^{2i}+z^{2i-1})$
$ B=sum sum b_{ij}(u_{i}otimes u_{j})$ ;;; which
$u_{i}(z)=sigma ^{-1/2}z^{2i-1}(alpha -z)$
Then we can use it to compute $langle Az^{j},z^{i}
angle $
explicity if A satisfies the previous condition.
ewline
Key words: shift operator, bounded matrix, dyadic recurrent formula,
slant Toeplitz operator, separable Hilbert space
| Identifer | oai:union.ndltd.org:NSYSU/oai:NSYSU:etd-0725107-195949 |
| Date | 25 July 2007 |
| Creators | Hsu, Chia-ming |
| Contributors | Jyh-Shyang Jeang, Ngai-Ching Wong, Mark C. Ho, Mu ming Wong |
| Publisher | NSYSU |
| Source Sets | NSYSU Electronic Thesis and Dissertation Archive |
| Language | English |
| Detected Language | English |
| Type | text |
| Format | application/pdf |
| Source | http://etd.lib.nsysu.edu.tw/ETD-db/ETD-search/view_etd?URN=etd-0725107-195949 |
| Rights | not_available, Copyright information available at source archive |
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