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We study the "quantized calculus" corresponding to the algebraic ideasrelated to "twisted cyclic cohomology" introduced in [KMT]. With very similardefinitions and techniques as those used in [jlo], we define and study "twistedentire cyclic cohomology" and the "twisted Chern character" associated with anappropriate operator-theoretic data called "twisted spectral data", whichconsists of a spectral triple in the conventional sense of noncommutativegeometry ([Con]) and an additional positive operator having specifiedproperties. Furthermore, it is shown that given a spectral triple (in theconventional sense) which is equivariant under the action of a compact matrixpseudogroup, it is possible to obtain a canonical twisted spectral data andhence the corresponding (twisted) Chern character, which will be invariantunder the action of the pseudogroup, in contrast to the fact that the Cherncharacter coming from the conventional noncommutative geometry need not beinvariant under the above action.

TWISTED ENTIRE CYCLIC COHOMOLOGY, J-L-O COCYCLES AND EQUIVARIANT SPECTRAL TRIPLES