I am interested in possible definitions, in particular to those which reduce, for k=1, to the standard formula f(A)=Q f(D) Q* where A=Q D Q* is the Schur decomposition (or spectral theorem). Some details are given in the attachment, also in connection with the beautiful applicative problem of defining correct geometric means of k positive definite matrices.
Consider functions f which can be approximated by polynomials. Then the question is how to generalize functions like f(a,b,c)=a^2 b c. A possible answer would be to symmetrize by averaging over all permutations, in my example, by all 24 permutations of a1, a2, b, c (a1 and a2 are just two copies of a). Probaby someone has studied this already...
The problem you seem not to address is that the product of two symmetric matrices will very often be not a symmetric matrix. As Richard Gill said, you must apply some kind of symmetrizing operation. And yes, the topic of evaluation of polynomials over non-commutative rings (like quaternions, to give a different example) is an often explored topic.
Yes, Viorel, I know. That's the point I'm making. If we knew how to do it for polynomials, we would know it for an awful lot more functions. How can we generalize a polynomial from R^k to R to a function from the set of k-tuples of Hermitian matrices to the set of Hermitian matrices?
Dear Stefano Serra-Capizzano,
Non commutative functional calculus in C*-algebra could help you. You can find a reference in "Nica & Speicher, lecture on the Combinatorics of free probability, Lecture 3".
Given matrices of size X_1, .., X_k, an elementary operation would be taking a *-polynomial in the matrices, that is considering a finite linear combination with complex coefficients or words in letters x_1, .., x_k and x*_1, .., x*_k (the empty word is the identity matrix), and replacing the x_j's by the X_j's.
The self adjoint seems constitutes elementary functions f for A_N the set of Hermitian matrices. If you complete the set of *-polynomials with your norm, you should obtain your generalization.
( I have no idea of what is Ando-Li-Mathias axioms)
In my opinion, the quetsion claimed above will sound reasonably if a really interesting and solid problem is formulated. to be solved. Then, one can figure out which really math structures are both essential and suitable for its solving and further their developing.
Regards!
What you are asking for is what people working in C^* algebra's call a functional calculus, i.e. a way to "evaluate" a complex or real function (of some kind), at an element of a C^* algebra (like your algebra of matrices) possibly assuming that the elements are hermitian or positive, or normal or... Now being able to "evaluate" a function is just a way to say that for every element u in the C^* algebra there is a *-algebra homomorphism from the space of functions to the C^* algebra is such a way that f(x) = x gets mapped to u. The canonical example is the functional calculus of the Hermitian operators on a Hilbert space. A Hermitian operator A can be "evaluated" on measurable functions f on the real line which are bounded on the spectrum of A using the spectral decomposition of the Hermitian operator. A general operator A can still be "evaluated" on the complex functions which are holomorphic on its spectrum using a variation of the Cauchy formula. If you don't know this already, see most any book on functional analysis (my favorite one, is the book by Rudin), and it is used extensively in quantum mechanics.
For the multivariate case you immediately run into two problems. For simplicity lets assume k = 2 so that we have to define an "evaluation" of two operators A and B. The easy problem is that we now have to distinguish two functions x and y such that x(A,B) = A and y(A,B) = B. We simply assume that the polynomial algebra R[x,y] is embedded as a sub algebra in our function space. The main problem is non commutativity. Clearly real (or complex) functions being commutative, the functional calculus must end in a commutative sub algebra of the (C^*)-algebra generated by A and B, and now we see that we cannot have our cake and eat it too: if A and B do not commute we cannot have an _algebra_ homomorphism from R[x,y] to the C^* algebra (in quantum mechanics this is known as the Van Hove theorem) so, a fortiori, cannot extend it to a function space. What we can try however is to take a nice sub algebra of R[x,y]. Now, as is well, known, the symmetric polynomials (in two variables) are polynomials in the elementary symmetric polynomials $\sigma_1$, $\sigma_2$ i.e. $R[x,y]^{S_2} = R[\sigma_1, sigma_2]$. Now we _can_ define a homomorphism $\phi_1$
R[\sigma_1, \sigma_2] \to C^*(A,B)^{S_2}
\sigma_1 \mapsto (A+B)
\sigma_2 \mapsto (A+B)^2
Now this is a bit disappointing because one would really like
\sigma_2 \mapsto (AB + BA)/2
but that does not work, because in general (A+B)(AB + BA) \ne (AB + BA)(A+B) !
There is however no problem to extend this homomorphism to continuous or even real measurable functions $f(\sigma_1, \sigma_2)$ of two variables, because for such a function one really has to "evaluate" the real function of one variable f(t, t^2) at $t = A+B$.
Also the homomorphism $\phi_1$ is far from unique if A and B are positive definite, we can define $\phi_2$
R[\sigma_1, \sigma_2] \to C^*(A,B)^{S_2}
\sigma_1 \mapsto A^{1/2}(A^{-1/2}BA^{-1/2})^{-1/2}A^{1/2}
\sigma_2 \mapsto \phi_2(\sigma_1)^2
and this extends to measurable functions for the same (cheap) reason as above.
At the end therefore one really has to be more precise about one wants to preserve.
I thank all the contributors, except 'spammers', so far for the pointers and the interesting suggestions.
Concerning the 'mathematical solid problem' mentioned by Prof. Anatolij Prykarpatski, it is true that it is required and, in fact, the motivation of my question concerns the caracterization of the geometric means of positive definite matrices, especially when they have some structure (say Toeplitzness, local Toeplitzness etc). Furtherore, the 10 Ando-Li-Mathias axioms are related to this problem (information for Dr. Camille Male)
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