Operator-Valued Functions 

Mathematics subject classification number:  47A56 

Maintained by Jean-Claude Evard. Last update: January 1, 2003

This page is a first draft at its early stage. It may already be useful to help 
communication between different research groups working on topics related to 
operators and matrices depending on variables or on matrices with entries in a 
ring of functions, in spite of the fact that an enormous amount of information is 
waiting to be added to this page.

Definitions.  An operator-valued function A is a function whose values A(t) 
are operators depending on a variable t. For each value of t,  A(t) is an 
operator from a Banach space E to a Banach space F. The variable t may be a 
real or a complex variable, or a vector t in a domain of a topological vector space, 
or an element t in a manifold. When t belongs to a finite dimensional space of 
dimension k, then t can be represented as an k-uple of variables: 
t = ( t1 , t2 , ... , tk ), and we say that the operator A(t) is a function of the 
k variables t1 , t2 , ... , tk . When the Banach spaces E and F are of finite 
dimensions n and m, the operator A(t) is an m by n matrix depending on t, and 
we say that the function A is a matrix-valued function. In this case, each entry of
the matrix A(t) is a scalar function aij( t ) of t, and the functions aij may belong 
to some given ring of scalar functions.

Motivation
In applied mathematics, it is exceptional that matrices and operators are constant. Most operators and 
matrices depend on variables. Such operators and matrices are called operator-valued functions and 
matrix-valued functions. Frequently, they appear in differential equations. To deal with such equations, 
it is necessary to: 
--- Extend calculus to non-commutative calculus with operator-valued functions.
--- Extend operator theory from constant operators to operators depending on variables.
--- Extend matrix theory from matrices with entries in a field to matrices with entries in a ring of functions.

The main directions of research about operator-valued functions are:
 1. Non-commutative calculus on operator-valued functions. 
2. Smooth reduction of an operator-valued function to canonical form. 
3. Algebraic approach to matrix-valued functions. 
4. Differential equations whose unknown function is an operator-valued function.
5. Algebraic approach to such equations.
6. Smooth solutions of algebraic equations whose unknown function is an operator-valued function.
=====================================================================
1. Non-commutative Calculus on operator-valued functions
This extension of Calculus is based on the theory of Lie Algebras. Examples of this extension can be 
found  in the thesis of M. J. Hellman [H] accepted in 1955, in the paper of W. Magnus [M] published in 
1954, and in the paper of A. D. Zieburg [Z] published in 1998.

2. Smooth reduction of an operator-valued function to canonical form
The most natural idea to reduce a differential equation whose unknown is an operator-valued function X 
to a simpler form is to reduce X to a simpler form. This raises the following fundamental problems.

Problem 1. Given a differentiable operator-valued function X defined on a manifold M, is it possible to 
find differentiable operator-valued functions S and J such that X(t) = S(t)J(t)S(t)^(-1) for every t in the 
manifold M, where J(t) is a canonical form of  X(t).

Problem 2. If the above reduction is not possible, under what condition does it exist a finite minimum 
number n of submanifolds M_1, M_2, ..., M_n of the manifold M together with n operator-valued
functions S_1, S_2, ..., S_n, such that for every t in the manifold M, there exist k in {1, 2, ..., n} 
such that X(t) = S_k(t)J(t)S_k(t)^(-1)?

     The differentiability of those operator-valued functions is essential for substitution into differential 
equations. This smooth reduction to canonical form  is based on bundle theory and algebraic topology.

2.1 Smooth bases of the kernel of an operator-valued function
     The smooth reduction of an operator-valued function to canonical form is based on the construction
of smooth bases of the kernel of an operator. For instance, for the differentiable reduction of a 
matrix-valued function A(t) to Jordan form, we have to construct a differentiable basis of 
Ker[A(t) - I]^m.

     A first fundamental paper on this subject was published by I.Z. Gochberg and J. Leiterer [GL] in 1976,
where conditions are established for the existence of bases of class C^p of the kernel and the image of an 
operator-valued functions defined on a compact manifold. This paper is based on the theory of cocycles. 
A extension of [GL] to the case where the domain of the operator-valued function is not compact has 
been obtained in [FGP] in the case where operators are matrices, using Bundle Theory. An extension 
of [FGP] from matrices to operators has been started in [DEG]. An approach without prerequisites is 
presented in [E3] for matrix-valued functions.

2.2 Smooth similarities of operator-valued functions
The general problem is given two differentiable operator-valued functions A and B that are 
defined on the same manifold M, and that are pointwise similar, that is A(t) = S(t)B(t)S(t)^(-1) for every t 
in M,  is it possible to find a differentiable operator-valued functin T such that A(t) = T(t)B(t)T(t)^(-1)
for every t in M. See [Gr] and [EG] for the case where the operators are matrices, and see the algebraic
approach for more general results.

3. Algebraic approach to the smooth similarities of matrix-valued functions
 This approach is based on the theory of matrices over a ring (each entry of a matrix-valued function 
belongs to a ring of functions); see [Gu 1-5].

4. Differential equations whose unknown function is an operator-valued function
As soon as we start to extend Calculus from scalar functions to operator-valued functions, we meet the problem of the commutation of an operator-valued function with its derivative. For instance, if X(t) is a square matrix depending on a real variable t, then by the Product Rule the derivative of X(t)^2 is X'(t)X(t)+X(t)X'(t), and this is equal to ordinary result 2X'(t)X(t) if and only if X(t) commutes with its derivative X'(t). Thus we immediately meet the problem of the study of the matrix differential equation X'(t)X(t)-X(t)X'(t) = 0. See [E1] and [E4].

5. Algebraic approach to differential equations whose 
    unknown function is an operator-valued function
 This approach is based on the algebraic approach to matrix-valued functions.
An example of this approach can be found in [AEG].

6. Smooth solutions of algebraic equations whose unknown 
    function is an operator-valued function.
See Section 9 of [E3].

References
[AEG] W. A. Adkins, J.-Cl. Evard, R. M. Guralnick, Matrices over differential fields which commute with their derivative,
           Linear Algebra Appl. 190: 253-261, 1993.
     [A]  V. I. Arnold, On matrices depending on parameters, Uspekhi Mat. Nauk 26, 1971.
[AEG]  Matrices over differential fields which commute with their derivative, with William Adkins and Robert Guralnick, 
            Linear Algebra and its Applications 190: 253-261, 1993.
[DEG]  Smooth parametrization of subspaces in a Banach space, M. Dupre, J.-Cl. Evard, and J. Glazebrook, Revista de la
            Union  Matematica Argentina 41 (2): 1-13, 1998.
   [E1]  On matrix functions which commute with their derivative, Linear Algebra and its Applications 68: 145-178, 1985.
   [E2]  Conditions for a vector subspace E(t) and for a projector P(t) not to depend on t, Linear Algebra and its Applications
            91: 121-131, 1987.
   [E3]  On the existence of bases of class C^p of the kernel and the image of a matrix function, Linear Algebra and its
            Applications 135: 33-67, 1990.
   [E4]  Invariance and commutativity properties of some classes of solutions of the matrix differential equation X(t)X'(t) = 
            X'(t)X(t), Linear Algebra and its Applications 218: 89-102, 1995.
  [EG]  On similarities of class C^p and applications to matrix differential equations, with Juan-Miguel Gracia, Linear Algebra
            and its Applications, special issue devoted to matrix-valued functions, 137: 363-386, 1990.
[FGP]  J. Ferrer, I. Garcia, and F. Puerta, Differentiable famillies of subspaces, Linear Algebra Appl. 199: 229-252 (1994).
   [Gr] J.-M. Gracia, Smooth Jordanization of conservative matrices, Report, CUA, 1985.
 [Gu1] R. Guralnick, A note on the local-global principle for similarity of matrices, Linear Algebra Appl. 30: 241-245, 1980.
 [Gu2] R. Guralnick, Similarity of matrices over local rings, Linear Algebra Appl. 41: 161-174, 1981.
 [Gu3] R. Guralnick, Matrix equivalence and isomorphism of modules, Linear Algebra Appl. 43: 125-136, 1982.
 [Gu4] R. Guralnick, Similarity of holomorphic matrices, Linear Algebra Appl. 99: 85-96, 1988.
 [Gu5] R. Guralnick, Similarity of matrices over commutative rings, Linear Algebra Appl. 157: 55-68, 1991. 
  [GL]  I.Z. Gochberg and J. Leiterer, Uber Algebren stetiger Operatorfunktionen, Studia Mathematica, LVII: 1-26, 1976.
     [H] M. J. Hellman, Lie algebras arising from systems of linear differential equations, Dissertation, Dept. of Mathematics,
           New York Univ., 1955.
    [M] W. Magnus, On the exponential solution of differential equations for a linear operator, Comm. Pure Appl. Math. VII:
           649-673, 1954.
     [R]  Haskell Rosenthal  Link , A version of the Jordan canonical form for general Banach spaces, 
            manuscript in progress, paper # 99,  Link , University of Texas at Austin. 
      [T] G. P. A. Thijsse, Global holomorphic similarity to a Jordan form, Results in Math. 8: 78-87, 1985.    
    [W] W. Wasow, On holomorphically similar matrices, J. Math. Anal. 4: 202-206, 1962.
      [Z] A.D. Ziebur, Chain rules for functions of matrices, Linear Algebra and its Applications, 283 (1-3): 87-97, 1998.

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