On the Connection Between the Indices of a Block Operator Matrix and of its Determinant

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.