Subadditivity, sublinearity, submultiplicativity, and other conditions are considered for spectra of pairs of operators on a Hilbert space. Sublinearity, for example, is a weakening of the well-known property~$L$ and means $\sigma(A+\lambda B) \subseteq \sigma(A) + \lambda \sigma(B)$ for all scalars $\lambda$. The effect of these conditions is examined on commutativity, reducibility, and triangularizability of multiplicative semigroups of operators. A sample result is that sublinearity of spectra implies simultaneous triangularizability for a semigroup of compact operators.

MSC Classifications:

47A15, 47D03, 15A30, 20A20, 47A10, 47B10 show english descriptions Invariant subspaces [See also 47A46]
Groups and semigroups of linear operators {For nonlinear operators, see 47H20; see also 20M20}
Algebraic systems of matrices [See also 16S50, 20Gxx, 20Hxx]
unknown classification 20A20
Spectrum, resolvent
Operators belonging to operator ideals (nuclear, $p$-summing, in the Schatten-von Neumann classes, etc.) [See also 47L20] 47A15 - Invariant subspaces [See also 47A46]
47D03 - Groups and semigroups of linear operators {For nonlinear operators, see 47H20; see also 20M20}
15A30 - Algebraic systems of matrices [See also 16S50, 20Gxx, 20Hxx]
20A20 - unknown classification 20A20
47A10 - Spectrum, resolvent
47B10 - Operators belonging to operator ideals (nuclear, $p$-summing, in the Schatten-von Neumann classes, etc.) [See also 47L20]

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