Algebras of Convolution Type Operators with Piecewise Slowly Oscillating Data. II: Local Spectra and Fredholmness

作者：Karlovich Yuri I^*; Loreto Hernandez Ivan

来源：Integral Equations and Operator Theory, 2013, 75(1): 49-86.

DOI：10.1007/s00020-012-2003-7

• convolution type operator
• Mellin pseudodifferential operator
• piecewise slowly oscillating function
• maximal ideal space
• limit operator
• spectra
• two idempotents theorem
• Fredholm symbol calculus

Let be the Banach algebra of all bounded linear operators acting on the weighted Lebesgue space , where and w is a Muckenhoupt weight. We study the Banach subalgebra of generated by all multiplication operators aI () and all convolution operators W (0)(b) (), where and are algebras of piecewise slowly oscillating functions that admit piecewise slowly oscillating discontinuities at arbitrary points of , and M (p,w) is the Banach algebra of Fourier multipliers on . Under some conditions on the Muckenhoupt weight w, using results of the local study of obtained in the first part of the paper and applying the theory of Mellin pseudodifferential operators and the two idempotents theorem, we now construct a Fredholm symbol calculus for the Banach algebra and establish a Fredholm criterion for the operators in terms of their Fredholm symbols. In four partial cases we obtain for more effective results.

• 出版日期2013-1