A Banach algebra A, a closed subalgebra B of A and an element a ∈ A such that sp(A, a) = sp(B, a).

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Let H be a Hilbert space, A = B(H), a = T be a nonzero element of A and also let B be a maximal commutative subalgebra containing T, then by Theorem 15.4 of [B&D, §15. Theorem 4],

sp(A, a) = sp(B, a).

Ref. [B&D] F.F. Bonsall, J. Duncan, Complete normed algebras, Springer-Verlag, 1973.