Even and odd operators on a graded Hilbert space

Definition

Let $H=H^0\oplus H^1$ be a graded Hilbert space, with grading operator $\Gamma$ . A bounded operator $T\in B(H)$ is even (of degree $0$ ) if $T(H^j)\subseteq H^j$ , equivalently $T\Gamma=\Gamma T$ . It is odd (of degree $1$ ) if $T(H^j)\subseteq H^{j+1\!\!\pmod 2}$ , equivalently $T\Gamma=-\Gamma T$ . An operator is homogeneous if it is even or odd, and its degree is written $|T|\in\mathbb Z/2$ . A general bounded operator need not be homogeneous, but decomposes uniquely as the sum of its even and odd parts.

Block-matrix form

Relative to $H^0\oplus H^1$ , the grading and the two homogeneous forms are

\Gamma= \begin{pmatrix} 1&0\\ 0&-1 \end{pmatrix}, \qquad T_{\mathrm{ev}}= \begin{pmatrix} T_{00}&0\\ 0&T_{11} \end{pmatrix}, \qquad T_{\mathrm{odd}}= \begin{pmatrix} 0&T_{01}\\ T_{10}&0 \end{pmatrix}.

For arbitrary $T\in B(H)$ , the projections onto these parts are

T_{\mathrm{ev}}=\frac12(T+\Gamma T\Gamma), \qquad T_{\mathrm{odd}}=\frac12(T-\Gamma T\Gamma).

Graded products and commutators

If $S$ and $T$ are homogeneous, then $ST$ has degree $|S|+|T|$ modulo $2$ . The graded commutator is

[S,T]_{\mathrm{gr}} =ST-(-1)^{|S||T|}TS.

Thus it is the ordinary commutator unless both operators are odd, in which case it is the anticommutator $ST+TS$ . This sign rule is the one used for graded representations, Fredholm modules, and Kasparov cycles.

Unbounded operators

Parity for an unbounded operator requires a domain condition. A densely defined operator $D$ is even or odd only when $\Gamma\operatorname{Dom}(D)=\operatorname{Dom}(D)$ and

D\Gamma=\Gamma D \quad\text{or}\quad D\Gamma=-\Gamma D

on $\operatorname{Dom}(D)$ , respectively. Bounded-operator formulas should not be applied to an unbounded $D$ without checking this invariant-domain hypothesis.

Convention

Some authors use graded operator for every operator on a graded space and homogeneous operator only for a pure degree. Here “even” and “odd” always assert homogeneity; no parity is assigned to a sum having both components.

References