Spelling suggestions: "subject:"plant toeplitz operator"" "subject:"plant teplitz operator""
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Operators which are constant with respect to slant Toeplitz operatorsChen, Chien-chou 04 July 2006 (has links)
Let H be a separable Hilbert space and {e_n : n belong to Z} be an orthonormal basis in H. A bounded operator T is called the slant Toeplitz operator if <T ej , ei> =c2i−j , where c_n is the n-th Fourier series of a bounded Lebesgue measurable function on the unit circle T = {z belong to C : |z| = 1}. It has been shown [7] that T* is an isometry if and only if |fi(z)|^2 +|fi(−z)|^2 = 2 a.e. on T and if this is the case and fi belong to C(T), then either T is unitarily equivalent to a shift or to the direct sum of a shift and a rank one unitary, with infinite multiplicity
(for the shift part, that is). Moreover, with some additional assumption on the smoothness and the zeros of fi, T* is similar to either the constant multiple of a shift or to the constant multiple of the direct sum of a shift and a rank one unitary, with infinite multiplicity. On the other hand, according to the terminologies in [10], an operator A that is constant with respect to a shift S if AS = SA and A S = SA . Therefore, in this article, we will study the operators that is constant with respect to T , i.e., bounded operator A satisfying AT = T A and A T = T A .
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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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On infinite matrices whose entries satisfying certain dyadic recurrent formulaHsu, Chia-ming 25 July 2007 (has links)
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
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