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Eigenvectors for Certain Action on B(H) Induced by ShiftCheng, Rong-Hang 05 September 2011 (has links)
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.
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Explicit Form of the Homogeneous Solutions for Some Operator EquationWang, Tsung-Chieh 20 January 2012 (has links)
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
$a,b,c,dinBbb C$.
par
It is clear that the solutions of the above system of equations
introduces a class of infinite matrices whose entries are related
``dyadically". In cite{Ho:g}, it is shown that the seemingly
complicated task of constructing these matrices can be carried out
alternatively in a systematical and relatively simple way by
applying the theory of Hardy classes of operators through certain
operator equation on ${cal B}({cal H})$ (space of bounded
operators on $cal H$) induced by a shift. Our purpose here is to
present explicit formula for the homogeneous solutions this equation.
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