View source for Multipliers

''of the first and second kinds''

The eigen values of the [[Monodromy operator|monodromy operator]] of a canonical equation.

In a complex Hilbert space, equations of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065540/m0655401.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065540/m0655402.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065540/m0655403.png" /> are self-adjoint operators, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065540/m0655404.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065540/m0655405.png" /> is periodic, are called canonical. In the finite-dimensional case the eigen values of the monodromy operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065540/m0655406.png" /> of this equation are called multipliers. If all solutions of a canonical equation are bounded on the entire real axis (the equation is stable), then the multipliers lie on the unit circle. Consider a canonical equation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065540/m0655407.png" /> with a real parameter <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065540/m0655408.png" />; then all multipliers can be divided into two groups: multipliers of the first (second) kind, which move counter-clockwise (clockwise) as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065540/m0655409.png" /> increases.

A canonical equation is called strongly stable if it is stable and remains stable under small variations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/m/m065/m065540/m06554010.png" />. For strong stability it is necessary and sufficient that all multipliers be on the unit circle and that there be no coincident multipliers of different kinds.

The theory of multipliers of the first and second kinds allows one to obtain a number of delicate tests for stability and estimates of the zone of stability for canonical equations. The homotopy classification of stable and unstable canonical equations has been given in terms of multipliers.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  Yu.L. Daletskii,   M.G. Krein,   "Stability of solutions of differential equations in Banach space" , Amer. Math. Soc.  (1974)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V.A. Yakubovich,   V.M. Starzhinskii,   "Linear differential equations with periodic coefficients" , Wiley  (1975)  (Translated from Russian)</TD></TR></table>



====Comments====


====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M.G. Krein,   "Topics in differential and integral equations and operator theory" , Birkhäuser  (1983)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  I. [I. Gokhberg] Gohberg,   P. Lancaster,   L. Rodman,   "Matrices and indefinite scalar products" , Birkhäuser  (1983)</TD></TR></table>