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Self-adjoint operators on inner product spaces over fields of power series

In: Mathematica Slovaca, vol. 58, no. 4
Hans A. Keller - Hermina A. Ochsenius

Details:

Year, pages: 2008, 455 - 482
Keywords:
self-adjoint operator, fields of power series, orthogonal decomposition, Pfister form
About article:
Theorems on orthogonal decompositions are a cornerstone in the classical theory of real (or complex) matrices and operators on $\Rn$. In the paper we consider finite dimensional inner product spaces $(E, Φ)$ over a field $K = F( (χ1 , …,χm ) )$ of generalized power series in $m$ variables and with coefficients in a real closed field $F$. It turns out that for most of these spaces $(E, Φ)$ every self-adjoint operator gives rise to an orthogonal decomposition of $E$ into invariant subspaces, but there are some salient exceptions. Our main theorem states that every self-adjoint operator $T : (E, Φ) \to (E, Φ)$ is decomposable except when $\dim E$ is a power of $2$ with exponent at most $m$, and $Φ$ is a tensor product of pairwise inequivalent binary forms. In the exceptional cases we provide an explicit description of indecomposable operators.
How to cite:
ISO 690:
Keller, H., Ochsenius, H. 2008. Self-adjoint operators on inner product spaces over fields of power series. In Mathematica Slovaca, vol. 58, no.4, pp. 455-482. 0139-9918.

APA:
Keller, H., Ochsenius, H. (2008). Self-adjoint operators on inner product spaces over fields of power series. Mathematica Slovaca, 58(4), 455-482. 0139-9918.