Keynote lecture 1: eigen-space relations amongst the universal matrices in fluctuation free approximation theory
Metin Demіralp · 2009
Fluctuation free approximations are used everywhere matrix representation is directly or indirectly involved. The idea is simple: The matrix representation of a function operator, whose action on its operand is the multiplication by a function, is equivalent to the image of the matrix representations of the independent variables appearing in the argument of the function, under that function when all terms are ignored. The terms are matrix representations of certain operators involving a operator which is called the fluctuation operator. This operator, in fact, projects its operand, which should be taken from an appropriately defined Hilbert space of functions, to the complement of an appropriately chosen subspace of this Hilbert space. The basis functions spanning the subspace are the members of a subset of the full basis function set which spans the mother Hilbert space. The matrix representations of the independent variables are defined with respect to these functions and therefore their number of rows or columns is equivalent to the dimension of the subspace under consideration. The basis functions span a space of multivariate functions. Therefore there should be more than one independent variable operators each of which multiplies its operand by a different independent variable. The matrix representations of these operators are called universal since they do not depend on the function mentioned in the free approximation. The independent variable operators naturally commute by definition. However this may not imply the commutativity amongst their matrix representations. The commutativity serves us to find a unique eigenfunction set to spectrally decompose the matrix representation of each independent variable such that the decompositions' projection matrices are constructed from this unique eigenfunction set while the linear combination coefficients vary from matrix representation to matrix representation. The case where the commutativity does not appear, one can conjecture that the norms of the commutators should decrease as the dimension of the subspace where the matrix representations are considered grows up to infinity. The operator appears once or more than once in the structure of the commutators and tends to go to zero as the subspace dimension increases. This is the reason why the commutators should diminish as the dimension of the considered Hilbert subspace grows unboundedly. All these urge us to investigate the eigenspaces of the matrices. Each of these spaces is one dimensional if the corresponding eigenvalue has no multiplicity. Even the case of multiple eigenvalues does not prevent to orthogonally decompose the corresponding eigenspaces to one dimensional ones because of the symmetry in the matrices. Each of one dimensional spaces related to eigenspaces characterizes one axis in the considered subspace. The axes systems should be peculiar to the related matrix unless all of them commute. That is, they do not coincide to form a unique coordinate system. However, the angles between the coordinate axes corresponding to different matrices should diminish as the dimension of the considered Hilbert subspace grows unboundedly. This speech focuses on the issues roughly mentioned above in details as much as possible and tries to make comments and remarks on the possible pitfalls and misunderstandings. The talk sufficiently addresses to the related works emphasizing on the findings of the author's and his group on this topics.