Elementary Quantum Field Theory by Ernest M. Henley, Walter Thirring

By Ernest M. Henley, Walter Thirring

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Show that if the product of two commuting elements of a unital algebra is invertible then both elements are invertible. 6. Let A be a Banach algebra with identity e and let a ∈ A be invertible. Show that a−1 ≥ r(a−1 ) = dist (σ (a), 0)−1 . 7. Consider the shift operator V : l 2 (Z+ ) → l 2 (Z+ ), (x0 , x1 , x2 , . ) → (0, x0 , x1 , x2 , . ). Show that V is a bounded operator and calculate its norm. Prove that V is left invertible but not right invertible, and determine its spectrum. 8. Let A be a unital Banach algebra, and let (an ) be a sequence in A with limit a ∈ A .

2 Invertibility and spectrum 27 that the functions defined by zn (t) := 1 − nt 0 if 0 ≤ t < 1/n, if 1/n ≤ t ≤ 1 ✷ fulfill the conditions of the definition. 26. Left (right) topological divisors of zero of a unital normed algebra cannot be invertible. Proof. Let B be a unital normed algebra and suppose z ∈ B to be a left topological divisor of zero. Then there exists a sequence (zn ) in B such that zn = 1 and zzn → 0. If z would be invertible, then z−1 zzn = zn → 0. But this is impossible because zn = 1 for all n ∈ N.

By hypothesis, l = e − ar is a left invertible element which lies in a maximal left ideal. 2. (i), (ii) ⇒ (iii): Let r ∈ RA and a ∈ A . Then e − ar is left invertible. We claim that it is right invertible, too. , (e + b)(e − ar) = e or, equivalently, b = (a + ba)r. Since r ∈ RA and the radical is a left ideal, we conclude that b belongs to the radical, too. Consequently, e + cb is left invertible for all c ∈ A . In particular, e + b is left invertible, and e + b is also right invertible by its definition.

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