  Using raising operators and geometric arguments, we establish formulas for
the K-theory classes of degeneracy loci in classical types. We also find new
determinantal and Pfaffian expressions for classical cases considered by
Giambelli: the loci where a generic matrix drops rank, and where a generic
symmetric or skew-symmetric matrix drops rank.
  In an appendix, we construct a K-theoretic Euler class for even-rank vector
bundles with quadratic form, refining the Chow-theoretic class introduced by
Edidin and Graham. We also establish a relation between top Chern classes of
maximal isotropic subbundles, which is used in proving the type D degeneracy
locus formulas.
