Let $l^2(Bbb Z)$ be the Hilbert space of square summable double sequences of complex numbers with standard basis ${e_n:ninBbb Z}$, and let us consider a bounded matrix $A$ on $l^2(Bbb Z)$
satisfying the following system of equations
egin{itemize}
item[1.] $lan
Ae_{2j},e_{2i}
an=p_{ij}+alan Ae_{j},e_i
an$;
item[2.] $lan
Ae_{2j},e_{2i-1}
an=q_{ij}+blan Ae_{j},e_{i}
an$;
item[3.] $lan
Ae_{2j-1},e_{2i}
an=v_{ij}+clan Ae_{j},e_{i}
an$;
item[4.] $lan
Ae_{2j-1},e_{2i-1}
an=w_{ij}+dlan Ae_{j},e_{i}
an$
end{itemize}
for all $i,j$, where $P=(p_{ij})$, $Q=(q_{ij})$, $V=(v_{ij})$, $W=(w_{ij})$ are bounded matrices on $l^2(Bbb Z)$ and $a,b,c,dinBbb C$. This type dyadic recurrent system arises in the study of bounded operators commuting with the slant Toeplitz operators, i.e., the class of operators ${{cal T}_vp:vpin L^infty(Bbb T)}$ satisfying $lan {cal T}_vp e_j,e_i
an=c_{2i-j}$, where $c_n$ is the $n$-th Fourier coefficient of $vp$.
It is shown in [10] that the solutions of the above system are closely related to the bounded solution $A$ for the operator equation
[
phi(A)=S^*AS=lambda A+B,
]
where $B$ is fixed, $lambdainBbb C$ and $S$ the shift given by ${cal T}_{arzeta+arxi z}^*$ (with $zetaxi
ot=0$ and $|zeta|^2+|xi|^2=1$). In this paper, we shall characterize the ``eigenvectors" for $phi$ for the eigenvalue $lambda$ with
$|lambda|leq1$, in terms of dyadic recurrent systems similar to the one above.
| Identifer | oai:union.ndltd.org:NSYSU/oai:NSYSU:etd-0905111-231911 |
| Date | 05 September 2011 |
| Creators | Cheng, Rong-Hang |
| Contributors | Jen-Chih Yao, Mark C. Ho, Jyh-shyang Jeang, Chun-Yen Chou |
| 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-0905111-231911 |
| Rights | unrestricted, Copyright information available at source archive |
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