• Source: Toeplitz operator

In operator theory, a Toeplitz operator is the compression of a multiplication operator on the circle to the Hardy space.




Details


Let




S

1




{\displaystyle S^{1}}

be the complex unit circle, with the standard Lebesgue measure, and




L

2


(

S

1


)


{\displaystyle L^{2}(S^{1})}

be the Hilbert space of square-integrable functions. A bounded measurable function



g


{\displaystyle g}

on




S

1




{\displaystyle S^{1}}

defines a multiplication operator




M

g




{\displaystyle M_{g}}

on




L

2


(

S

1


)


{\displaystyle L^{2}(S^{1})}

. Let



P


{\displaystyle P}

be the projection from




L

2


(

S

1


)


{\displaystyle L^{2}(S^{1})}

onto the Hardy space




H

2




{\displaystyle H^{2}}

. The Toeplitz operator with symbol



g


{\displaystyle g}

is defined by





T

g


=
P

M

g



|


H

2




,


{\displaystyle T_{g}=PM_{g}\vert _{H^{2}},}


where " | " means restriction.
A bounded operator on




H

2




{\displaystyle H^{2}}

is Toeplitz if and only if its matrix representation, in the basis



{

z

n


,
z
∈

C

,
n
≥
0
}


{\displaystyle \{z^{n},z\in \mathbb {C} ,n\geq 0\}}

, has constant diagonals.




Theorems


Theorem: If



g


{\displaystyle g}

is continuous, then




T

g


−
λ


{\displaystyle T_{g}-\lambda }

is Fredholm if and only if



λ


{\displaystyle \lambda }

is not in the set



g
(

S

1


)


{\displaystyle g(S^{1})}

. If it is Fredholm, its index is minus the winding number of the curve traced out by



g


{\displaystyle g}

with respect to the origin.
For a proof, see Douglas (1972, p.185). He attributes the theorem to Mark Krein, Harold Widom, and Allen Devinatz. This can be thought of as an important special case of the Atiyah-Singer index theorem.

Axler-Chang-Sarason Theorem: The operator




T

f



T

g


−

T

f
g




{\displaystyle T_{f}T_{g}-T_{fg}}

is compact if and only if




H

∞


[



f
¯



]
∩

H

∞


[
g
]
⊆

H

∞


+

C

0


(

S

1


)


{\displaystyle H^{\infty }[{\bar {f}}]\cap H^{\infty }[g]\subseteq H^{\infty }+C^{0}(S^{1})}

.
Here,




H

∞




{\displaystyle H^{\infty }}

denotes the closed subalgebra of




L

∞


(

S

1


)


{\displaystyle L^{\infty }(S^{1})}

of analytic functions (functions with vanishing negative Fourier coefficients),




H

∞


[
f
]


{\displaystyle H^{\infty }[f]}

is the closed subalgebra of




L

∞


(

S

1


)


{\displaystyle L^{\infty }(S^{1})}

generated by



f


{\displaystyle f}

and




H

∞




{\displaystyle H^{\infty }}

, and




C

0


(

S

1


)


{\displaystyle C^{0}(S^{1})}

is the space (as an algebraic set) of continuous functions on the circle. See S.Axler, S-Y. Chang, D. Sarason (1978).




See also


Toeplitz matrix – Matrix with shifting rows




References



S.Axler, S-Y. Chang, D. Sarason (1978), "Products of Toeplitz operators", Integral Equations and Operator Theory, 1 (3): 285–309, doi:10.1007/BF01682841, S2CID 120610368{{citation}}: CS1 maint: multiple names: authors list (link)
Böttcher, Albrecht; Grudsky, Sergei M. (2000), Toeplitz Matrices, Asymptotic Linear Algebra, and Functional Analysis, Birkhäuser, ISBN 978-3-0348-8395-5.
Böttcher, A.; Silbermann, B. (2006), Analysis of Toeplitz Operators, Springer Monographs in Mathematics (2nd ed.), Springer-Verlag, ISBN 978-3-540-32434-8.
Douglas, Ronald (1972), Banach Algebra techniques in Operator theory, Academic Press.
Rosenblum, Marvin; Rovnyak, James (1985), Hardy Classes and Operator Theory, Oxford University Press. Reprinted by Dover Publications, 1997, ISBN 978-0-486-69536-5.

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