By Pietro Aiena

Indicates the interaction among Fredholm thought and native spectral concept. For graduate classes in operator concept and mathematicians.

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Extra info for Fredholm and Local Spectral Theory, with Applications to Multipliers

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70. For every bounded operator T ∈ L(X), X a Banach space, we have: (i) H0 (T ) ⊆⊥ K(T ⋆ ) and K(T ) ⊆⊥ H0 (T ⋆ ); (ii) If T is essentially semi-regular, then (24) H0 (T ) = N ∞ (T ) =⊥ K(T ⋆ ) and K(T ) =⊥ H0 (T ⋆ ). In particular, these equalities hold for every T ∈ Φ± (X), or for every semiregular operator. (iii) If T is semi-regular then H0 (T ) ⊆ K(T ). Proof (i) Consider an element u ∈ H0 (T ) and f ∈ K(T ⋆ ). From the definition of K(T ⋆ ) we know that there exists δ > 0 and a sequence (gn ), n ∈ Z+ of X ⋆ such that g0 = f, T ⋆ gn+1 = gn and gn ≤ δ n f for every n ∈ Z+ .

47. Let T ∈ L(X), X a Banach space. Suppose that either of the two following conditions hold: (i) N ∞ (T ) ⊂e T ∞ (X); (ii) N ∞ (T ) ⊂e T (X). Then for every n, m ∈ N we have ker T n ⊂e T m (X). Proof Suppose that the inclusion (i) holds, namely, there is a finitedimensional space F for which N ∞ (T ) ⊆ T ∞ (X) + F . Since ker T n ⊆ N ∞ (T ) and T m (X) ⊇ T ∞ (X), for all n, m ∈ N, we easily obtain that ker T n ⊆ T m (X) + F , for every n, m ∈ N. Assume that the inclusion (ii) is fulfilled. Here we proceed by induction on m.

This implies that there exists a sequence (xn ) with xn ∈ xn such that dist (xn − x, ker (T )), the distance of xn − x to ker T , converges to zero, and so there exists a sequence (zn ) ⊂ ker T such that xn − x − zn → 0. Then the sequence (xn − zn ) ⊂ Y + ker T converges to x and since by assumption Y + ker T is closed, we have x ∈ Y + ker T . This implies x ∈ Y ; thus Y is closed . The next result shows that the semi-regularity of an operator may be characterized in terms of the continuity of certain maps.

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Fredholm and Local Spectral Theory, with Applications to by Pietro Aiena
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