A framework for solving various positive-definite and strictly contractive extension problems and various interpolation problems from a unified abstract point of view. This method applies to algebras with band structure. An algebra with an identity and an involution is called an algebra with band structure if admits a direct sum decomposition
where all the summands are subspaces of such that the following conditions are satisfied:
) , , ;
) the following multiplication table holds:
The space is called the band of , and any element in is called a diagonal. Also, let
The natural projections associated with the decomposition (a1) are denoted by , respectively.
An example of an algebra with band structure is the Wiener algebra of all complex-valued functions on the unit circle that have absolutely convergent Fourier series expansions
The involution on is complex conjugation. Let be a fixed positive integer. A band structure on is obtained by letting the summands in (a1) be defined by
An element in an algebra with involution and unit is called positive definite in if for some invertible . Such an element in an algebra with band structure is said to admit a right (respectively, left) spectral factorization if and can be taken in (respectively, ).
Hereafter, is assumed to be a -subalgebra of a unital -algebra , with the unit of equal to the unit of . Let be an element in the band . An -positive extension of is an element that is positive definite in such that
for certain elements and . A band extension of is an -positive extension of such that . The main problems are to determine conditions under which a band extension of exists, to find the band extension when it exists, and to describe all -positive extensions of when has a band extension. The following two statements provide solutions of these problems.
I) Let be an algebra with band structure (a1), and let . Then has a band extension with a right spectral factorization relative to (a1) if and only if the equation
has a solution with the following properties:
ii) is invertible and ;
iii) for some which is invertible in . Furthermore, in this case such an element is obtained by taking
where is any solution of (a2) satisfying i)–iii).
To describe all -positive extensions of , it will be additionally assumed that the following axiom holds:
Axiom : If and , then .
This axiom holds if is closed in .
II) Let be an algebra with band structure (a1) in a unital -algebra , and assume that axiom holds. Let , and suppose that has a band extension which admits a right and left spectral factorization relative to (a1):
Then each -positive extension of is of the form
where the free parameter is an arbitrary element in such that . Moreover, the mapping provides a one-to-one correspondence between all such and all -positive extensions of .
In the above statement, may be replaced by
where now the free parameter is an arbitrary element of such that .
The right-hand side of (a3) yields a positive extension (i.e., an extension which is positive definite in ) if and only if the free parameter is such that is positive definite in .
An alternative characterization of the band extension is provided by an abstract maximum entropy principle. For this it is necessary to assume two additional axioms. An element is positive semi-definite in if for some .
Axiom : If is positive semi-definite in , then is positive semi-definite in .
Axiom : If is positive semi-definite in and , then .
Any element of with a right spectral factorization can be factored uniquely in the form
where and is invertible with . The element is called the right multiplicative diagonal of and is denoted by . The maximum entropy principle states that if a self-adjoint element in (cf. also Self-adjoint operator) has a band extension with a right spectral factorization, then for any -positive extension of having a right spectral factorization,
with equality only if .
Solution of the Carathéodory–Toeplitz extension problem.
There are many applications of these results to various algebras of functions, matrix-valued functions, and matrices. When applied to the Wiener algebra with the band structure described above, they yield a description of the solutions of the classical Carathéodory–Toeplitz extension problem.
Given a trigonometric polynomial
one looks for a function in with the property that for every . The following statement gives the solution.
The Carathéodory–Toeplitz extension problem for the trigonometric polynomial (a4) is solvable if and only if the matrix
is positive definite. In that case there exists a unique solution with the additional property that the th Fourier coefficient of is equal to for .
To obtain , let
Then for and for and
Furthermore, every solution of the Carathéodory–Toeplitz problem is of the form
where is an arbitrary function with for and with the th Fourier coefficient of equal to for . Moreover, (a5) gives a one-to-one correspondence between all such and all solutions . Additionally, the band solution is the unique solution that maximizes the entropy integral
This solution is called the maximum entropy solution.
|[a1]||I. Gohberg, S. Goldberg, M.A. Kaashoek, "Classes of linear operators II" , Operator Theory: Advances and Applications , 63 , Birkhäuser (1993)|
|[a2]||H. Dym, I. Gohberg, "Extensions of kernels of Fredholm operators" J. Anal. Math. , 42 (1982/3) pp. 51–97|
|[a3]||H. Dym, I. Gohberg, "A new class of contractive interpolants and maximum entropy principles" , Operator Theory: Advances and Applications , 29 , Birkhäuser (1988) pp. 117–150|
|[a4]||I. Gohberg, M.A. Kaashoek, H.J. Woerdeman, "The band method for positive and conntractive extension problems" J. Operator Th. , 22 (1989) pp. 109–155|
|[a5]||I. Gohberg, M.A. Kaashoek, H.J. Woerdeman, "The band method for positive and conntractive extension problems: An alternative version and new applications" Integral Eq. Operator Th. , 12 (1989) pp. 343–382|
|[a6]||I. Gohberg, M.A. Kaashoek, H.J. Woerdeman, "A maximum entropy principle in the general framework of the band method" J. Funct. Anal. , 95 (1991) pp. 231–254|
Band method. Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Band_method&oldid=42683