On the semi-Browder spectrum

An operator in a Banach space is called upper (lower) semi-Browder if it is upper (lower) semi-Fredholm and has a finite ascent (descent). We extend this notion to n-tuples of commuting operators and show that this notion defines a joint spectrum. Further we study relations between semi-Browder and (essentially) semiregular operators. Denote by L(X) the algebra of all bounded linear operators in a complex Banach space X and by I the identity operator in X. For T in L(X) denote by N(T ) = {x ∈ X : Tx = 0} and R(T ) = {Tx : x ∈ X} its kernel and range, respectively. Denote further R∞(T ) = ⋂∞ k=0 R(T ) and N∞(T ) = ⋃∞ k=0 N(T ). The sets of all upper (lower) semi-Fredholm operators in X will be denoted by Φ+(X) and Φ−(X). Recall that T ∈ Φ+(X) if and only if dim N(T ) < ∞ and R(T ) is closed; T ∈ Φ−(X) if and only if codim R(T ) < ∞ (then R(T ) is closed automatically). The ascent and descent of T are defined by a(T ) = min{n : N(T) = N(Tn+1)} and d(T ) = min{n : R(T) = R(Tn+1)}. We say that an operator T ∈ L(X) is upper (lower) semi-Browder if it is upper (lower) semi-Fredholm and has a finite ascent (descent). The set of all upper (lower) semi-Browder operators in X will be denoted by B+(X) and B−(X). Semi-Browder operators were studied by many authors, see e.g. [4], [12], [14], [18], [20], [21], [22], [24]. The name was introduced in [6]. We extend the notion of semi-Browder operators to n-tuples of commuting operators. We discuss the lower semi-Browder case; the upper case is dual. Let T = (T1, ..., Tn) be an n-tuple of mutually commuting operators in a Banach space X. We use the standard multiindex notation. Denote by Z+ the set of all non-negative integers. If α = (α1, ..., αn) ∈ Z+ then denote |α| = α1 + · · · + αn and T = T1 1 · · ·Tαn n . For k = 0, 1, 2, ..., denote Mk(T ) = R(T k 1 ) + · · · + R(T k n ) and let M ′ k(T ) be the smallest subspace of X containing the set ⋃{R(Tα) : α ∈ Z + and |α| = k}. Clearly X = M0(T ) ⊃ M1(T ) ⊃ M2(T ) ⊃ · · · and X = M ′ 0(T ) ⊃ M ′ 1(T ) ⊃ M ′ 2(T ) ⊃ · · ·. Further M ′ n(k−1)+1(T ) ⊂ Mk(T ) ⊂ M ′ k(T ). (1) Indeed, if α = (α1, ..., αn) ∈ Z + and |α| = n(k − 1) + 1 then there exists i, 1 ≤ i ≤ n such that αi ≥ k, so that R(T) ⊂ R(T k i ) ⊂ Mk(T ). This proves the first inclusion of (1) and the second inclusion is clear. Denote R∞(T ) = ⋂∞ k=0 Mk(T ) = ⋂∞ k=0 M ′ k(T ). AMS Subject Classification (1991): 47A10, 47A53, 47A55 The first two authors were supported by the grant No. 119106 of the Academy of Sciences, Czech Republic. The work of the third-named author was supported by the Science Fund of Serbia, grant number 04M03, through Matematički Institut.