An Algebra of Pseudodifferential Operators with Slowly Oscillating Symbols

Let VR denote the Banach algebra of absolutely continuous functionsof bounded total variation on R, and let B_p be the Banach algebraof bounded linear operators acting on the Lebesgue space L^pRfor 1 < p < . We study the Banach algebra A B_p generatedby the pseudodifferential operators of zero order with slowlyoscillating VR-valued symbols on R. Boundedness and compactnessconditions for pseudodifferential operators with symbols inL R, VR are obtained. A symbol calculus for the non-closed algebraof pseudodifferential operators with slowly oscillating VR-valuedsymbols is constructed on the basis of an appropriate approximationof symbols by infinitely differentiable ones and by use of thetechniques of oscillatory integrals. As a result, the quotientBanach algebra A = A K, where K is the ideal of compact operatorsin B_p, is commutative and involutive. An isomorphism betweenthe quotient Banach algebra A of pseudodifferential operatorsand the Banach algebra of their Fredholm symbols is established. A Fredholm criterionand an index formula for the pseudodifferential operators A A are obtained in terms of their Fredholm symbols. 2000 MathematicsSubject Classification 47G30, 47L15 (primary), 47A53, 47G10(secondary).