The Weyl–von Neumann theorem for skew-symmetric operators
Tóm tắt
We consider the diagonalization of self-adjoint operators in the infinite-dimensional orthogonal Lie algebra
$$\mathcal O_C$$
, which consists of all bounded linear operators T on a separable, infinite-dimensional, complex Hilbert space
$$\mathcal {H}$$
satisfying
$$CTC = -T^*$$
, where C is a conjugation on
$$\mathcal H$$
. First, we establish the Weyl–von Neumann Theorem in
$$\mathcal {O}_C$$
, showing that each self-adjoint operator in
$$\mathcal {O}_C$$
can be diagonalized in
$$\mathcal {O}_C$$
up to arbitrarily small perturbations with respect to the Schatten p-norm,
$$p\in (1,\infty )$$
. Second, for any
$$n\ge 2$$
, we prove that n commuting self-adjoint operators in
$$\mathcal {O}_C$$
can be diagonalized simultaneously in
$$\mathcal {O}_C$$
up to arbitrarily small perturbations with respect to the Schatten n-norm. This is an
$$\mathcal {O}_C$$
-analogue of Voiculescu’s result concerning the simultaneous diagonalization of commuting self-adjoint operators. Finally, as an application of the preceding results, we show that those irreducible ones constitute a norm dense subset of
$$\mathcal {O}_C$$
.
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