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# Algebra lineal y algunas de sus aplicaciones by L.I. GOLOVINA By L.I. GOLOVINA

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I − 1. Likewise, if we wish to study etA x for a given x ∈ X, we use a basis BA,x of X containing the non-zero elements of BA,x := (A − λi )ν x : i = 1, . . , m; ν = 0, . . , νi − 1 . ν! 5). The behavior of a function gν,λ (t) := tν etλ is easy to understand and essentially depends on the real part of λ. The following cases are possible. • Re λ < 0. Then, for each ﬁxed value of ν, etλ tν → 0 as t → ∞, where the decay is exponential in the following sense: for any 0 < δ < − Re λ there is Mδ ≥ 1 such that eλt tν = tν etReλ ≤ Mδ e−δt for all t ≥ 0.

For T ∈ L(X) the following assertions are equivalent. (i) (ii) (iii) (iv) (T k ) is Ces` aro summable. limk→∞ k −1 T k = 0. (T k ) is bounded. r(T ) ≤ 1 and each eigenvalue of modulus 1 is a simple pole of the resolvent. In any of these equivalent cases the sequence T (k) of Ces` aro means of T converges to 0 if 1 ∈ / σ(T ), and to the spectral projection P1 belonging to 1 if 1 ∈ σ(T ). Proof. (i) =⇒ (ii) follows since we can write T k = kT (k) − (k − 1)T (k−1) . b). It remains to show (iv) =⇒ (i) and the assertion on the limit of the Ces`aro means.

A − λj , λi − λj i = 1, . . , m. a) Determine the eigenvalues λi and the corresponding multiplicities νi for the matrices ⎛ ⎞ 1 0 0 0 0 A= and B = ⎝0 1 0⎠ . 0 1 0 1 1 b) Discuss further matrices you ﬁnd interesting. 30 Chapter 2. Functional Calculus 6 3 4. Calculate etA , An , and sin(tA) for A = −1 . Discuss further matrices 2 you ﬁnd interesting. ∞ 5. Show that if B = S −1 AS, where S is an invertible matrix, and if f ∈ CA , ∞ then f ∈ CB and f (B) = S −1 f (A)S. 6. , in the situation of X = Rn , L(X) = Mn (R), and R[x]).