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On a class of generalized Fredholm operators, VI

Schmoeger C.

Demonstratio Mathematica

1999

Vol. 32, nr 4

811-822

Let X be a complex Banach space and T a generalized Fredholm operator on X (see [3] , [4] , [5] , [6] and [7] ). In [7] we have shown that T has a Kato decomposition (Xl, X2). We say that a Kato decomposition (Xl, X2) of T is non-trivial if X2= {0}. The main result of this paper reads as follows: Let T be a generalized Fredholm operator with a non-trivial Kato decomposition. Then (i) The subspace X2 of each Kato decomposition of T is unique if and only if T has finite ascent. (ii) The subspace Xl of each Kato decomposition of T is unique if and only if T has finite descent. (iii) T has a unique Kato decomposition if and only if 0 is a pole of the resolvent (T - lambaI)-l.

On a class of generalized Fredholm operators, V

Schmoeger C.

Demonstratio Mathematica

1999

Vol. 32, nr 3

595-604

Let X be a complex Banach space and T a bounded linear operator on X. T is called a generalized Fredholm operator if T is relatively regular and if for some pseudo-inverse S of T the operator I - ST - T is Fredholm. The main result of this paper reads as follows: T is a generalized Fredholm operator if and only if T = T1 T2 , whereT1 is a Fredholm operator with jump j(Tl ) = 0 and T2 is a finite-dimensional nilpotent operator.

On a class of generalized Fredholm operators, IV

Schmoeger C.

Demonstratio Mathematica

1999

Vol. 32, nr 3

581-594

In the present paper we investigate generalized Fredholm operators (see [15], t16] and [17]) on a complex Hilbert space H. The main results of this paper read as follows.