- ON A PROBLEM OF HALMOS ABOUT INVERTIBLE OPERATORS
- Hwang, J.S. ;
- Bulletin of the Korean Mathematical Society, volume 24, issue 1, 1987, Pages 27~29
Abstract
If A is a bounded linear operator on a Hilbert space, and if the operator norm ∥I-A∥<1, then A is invertible, see Halmos [2, pp.52]. That assertion, not only for operators, on Hillbert spaces, but for arbitrary elements of Banach algebras, is usually proved by consideration of the infinite series (Fig.) see Dunford and Schwarts [1, pp.585]. In this operation, the completeness is sufficient. Recently, in [3], Halmos asked the question as to whether the assertion is true without completeness\ulcorner In other words, is there a bounded linear operator A on an inner product space, such that ∥I-A∥<1 but A is not invertible\ulcorner