$G_{2n}$ spaces

Abstract:

A complex normed linear space $X$ will be called a ${G_{2n}}$ space if and only if there is a mapping $\left \langle { \cdot , \ldots , \cdot } \right \rangle$ from ${X^{2n}}$ into the complex numbers such that: ${x_k} \to \left \langle {{x_1}, \ldots ,{x_{2n}}} \right \rangle$ is linear for $k = 1, \ldots ,n;\left \langle {{x_1}, \ldots ,{x_{2n}}} \right \rangle = {\left \langle {{x_{2n}}, \ldots ,{x_1}} \right \rangle ^ - }$; and ${\left \langle {x, \ldots ,x} \right \rangle ^{1/2n}} = ||x||$. The basic models are the ${L^{2n}}$ spaces, but one also has that every inner product space is a ${G_{2n}}$ space for every integer $n$. Hence ${G_{2n}}$ spaces of a given cardinality need not be isometrically isomorphic. It is shown that a complex normed linear space is a ${G_{2n}}$ space if and only if the norm satisfies a generalized parallelogram law. From the proof of this characterization it follows that a linear map $U$ from $X$ to $X$ is an isometry if and only if $\left \langle {U({x_1}), \ldots ,U({x_{2n}})} \right \rangle = \left \langle {{x_1}, \ldots ,{x_{2n}}} \right \rangle$ for all ${x_1}, \ldots ,{x_{2n}}$. This then provides a way to construct all of the isometries of a finite dimensional ${G_{2n}}$ space. Of particular interest are the $\operatorname {CBS} {G_{2n}}$ spaces in which $|\left \langle {{x_1}, \ldots ,{x_{2n}}} \right \rangle | \leqq ||{x_1}|| \cdots ||{x_{2n}}||$. These spaces have many properties similar to inner product spaces. An operator $A$ on a complete $\operatorname {CBS} {G_{2n}}$ space is said to be symmetric if and only if $\left \langle {{x_1}, \ldots ,A({x_i}), \ldots ,{x_{2n}}} \right \rangle = \left \langle {{x_1}, \ldots ,A({x_j}), \ldots ,{x_{2n}}} \right \rangle$ for all $i$ and $j$. It is easy to show that these operators are scalar and that on ${L^{2n}},n > 1$, they characterize multiplication by a real ${L^\infty }$ function. The interest in nontrivial symmetric operators is that they exist if and only if the space can be decomposed into the direct sum of nontrivial ${G_{2n}}$ spaces.