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- W714942 abstract "We consider a finite block operator matrix $$ mathcal{A}$$ in a Hilbert space. If the entries of $$ mathcal{A}$$ commute modulo the compact operators, then $$ mathcal{A}$$ is a Fredholm operator if and only if det $$ mathcal{A}$$ is a Fredholm operator, but in general ind $$ mathcal{A}$$ ≠ ind det $$ mathcal{A}$$ . On the other hand, if the commutators of the entries of $$ mathcal{A}$$ are trace class operators then ind $$ mathcal{A}$$ = ind det $$ mathcal{A}$$ . We obtain formulas for the difference ind $$ mathcal{A}$$ — ind det $$ mathcal{A}$$ provided the entries of $$ mathcal{A}$$ commute modulo some von Neumann—Schatten ideal. Then we indicate some ideals larger than the ideal of trace class operators for which the mentioned statement about the equality ind $$ mathcal{A}$$ = ind det $$ mathcal{A}$$ remains true." @default.
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- W714942 date "2006-01-01" @default.
- W714942 modified "2023-10-18" @default.
- W714942 title "On the Connection Between the Indices of a Block Operator Matrix and of its Determinant" @default.
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- W714942 doi "https://doi.org/10.1007/978-3-7643-7737-3_6" @default.
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