## Tokyo Journal of Mathematics

### The Maslov Index: a Functional Analytical Definition and the Spectral Flow Formula

#### Abstract

We give a functional analytical definition of the Maslov index for continuous curves in the Fredholm-Lagrangian Grassmannian. Our definition does not require assumptions either at the endpoints or at the crossings of the curve with the Maslov cycle. We demonstrate an application of our definition by developing the symplectic geometry of self-adjoint extensions of unbounded symmetric operators. We discuss continuous variations of the form $A_D+C_t$, where $A_D$ is a fixed self-adjoint unbounded Fredholm operator and $\{C_t\}$ a family of bounded self-adjoint operators. We extend the definition of the spectral flow to such families of unbounded operators in a purely functional analytical way. We then prove that the spectral flow is equal to the Maslov index of the corresponding family of abstract Cauchy data spaces.

#### Article information

Source
Tokyo J. Math., Volume 21, Number 1 (1998), 1-34.

Dates
First available in Project Euclid: 31 March 2010

https://projecteuclid.org/euclid.tjm/1270041982

Digital Object Identifier
doi:10.3836/tjm/1270041982

Mathematical Reviews number (MathSciNet)
MR1630119

Zentralblatt MATH identifier
0932.37063

#### Citation

BOOSS-BAVNBEK, Bernhelm; FURUTANI, Kenro. The Maslov Index: a Functional Analytical Definition and the Spectral Flow Formula. Tokyo J. Math. 21 (1998), no. 1, 1--34. doi:10.3836/tjm/1270041982. https://projecteuclid.org/euclid.tjm/1270041982