A Banach algebra A Banach algebra A, a closed subalgebra B of A and an element a Î 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.

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