Some representation theorems for sesquilinear forms

A Primer on Sesquilinear Forms

The dot product is an important tool for calculations in Rn. For instance, we can use it to measure distance and angles. However, it doesn’t come from just the vector space structure on Rn – to define it implicitly involves a choice of basis. In Math 67, you may have studied inner products on real vector spaces and learned that this is the general context in which the dot product arises. Now, w...

Some Representation Theorems for Partially Ordered Sets

Canonical matrices of bilinear and sesquilinear forms

Canonical matrices are given for • bilinear forms over an algebraically closed or real closed field; • sesquilinear forms over an algebraically closed field and over real quaternions with any nonidentity involution; and • sesquilinear forms over a field F of characteristic different from 2 with involution (possibly, the identity) up to classification of Hermitian forms over finite extensions of...

Reverse Cauchy–schwarz Inequalities for Positive C∗-valued Sesquilinear Forms

We prove two new reverse Cauchy–Schwarz inequalities of additive and multiplicative types in a space equipped with a positive sesquilinear form with values in a C *-algebra. We apply our results to get some norm and integral inequalities. As a consequence, we improve a celebrated reverse Cauchy–Schwarz inequality due to G. Pólya and G. Szegö.

Sesquilinear forms over rings with involution

Many classical results concerning quadratic forms have been extended to Hermitian forms over algebras with involution. However, not much is known in the case of sesquilinear formswithout any symmetry property. The present paperwill establish aWitt cancellation result, an analogue of Springer’s theorem, as well as some local–global and finiteness results in this context. © 2013 Elsevier B.V. All...