Paranormal operator on a Hilbert space

In this thesis an extensive study is made of the set <i>Pi> of all paranormal operators in B(<i>Hi>), the set of all bounded endomorphisms on the complex Hilbert space <i>Hi>. T ϵ B(<i>Hi>) is paranormal if for each z contained in the resolvent set of T, d(z, σ(T))//(T-zI)^-1 = 1 where d(z, σ(T)) is the distance from z to σ(T), the spectrum of T. <i>Pi> contains the set <i>Ni> of normal operators and <i>Pi> contains the set of hyponormal operators. However, <i>Pi> is contained in <i>Li>, the set of all T ϵ B(<i>Hi>) such that the convex hull of the spectrum of T is equal to the closure of the numerical range of T. Thus, <i>Ni>≤<i>Pi>≤<i>Li>.

If the uniform operator (norm) topology is placed on B(<i>Hi>), then the relative topological properties of <i>Ni>, <i>Pi>, <i>Li> can be discussed. In Section IV, it is shown that: 1) <i>N P i> and <i>Li> are arc-wise connected and closed, 2) <i>N, P,i> and <i>Li> are nowhere dense subsets of B(<i>Hi>) when dim <i>Hi> ≥ 2, 3) <i>Ni> = <i>Pi> when dim<i>Hi> ˂ ∞ , 4) <i>Ni> is a nowhere dense subset of <i>Pi> when dim<i>Hi> ˂ ∞ , 5) <i>Pi> is not a nowhere dense subset of <i>Li> when dim<i>Hi> ˂ ∞ , and 6) it is not known if <i>Pi> is a nowhere dense subset of <i>Li> when dim<i>Hi> ˂ ∞.

The spectral properties of paranormal operators are of current interest in the literature. Putnam [22, 23] has shown that certain points on the boundary of the spectrum of a paranormal operator are either normal eigenvalues or normal approximate eigenvalues. Stampfli [26] has shown that a hyponormal operator with countable spectrum is normal. However, in Theorem 3.3, it is shown that a paranormal operator T with countable spectrum can be written as the direct sum, N ⊕ A, of a normal operator N with σ(N) = σ(T) and of an operator A with σ(A) a subset of the derived set of σ(T). It is then shown that A need not be normal. If we restrict the countable spectrum of T ϵ <i>Pi> to lie on a C²-smooth rectifiable Jordan curve Gb>ob>, then T must be normal [see Theorem 3.5 and its Corollary]. If T is a scalar paranormal operator with countable spectrum, then in order to conclude that T is normal the condition of σ(T) ≤ Gb>ob> can be relaxed [see Theorem 3.6]. In Theorem 3.7 it is then shown that the above result is not true when T is not assumed to be scalar. It was then conjectured that if T ϵ <i>Pi> with σ(T) ≤ Gb>ob>, then T is normal. The proof of Theorem 3.5 relies heavily on the assumption that T has countable spectrum and cannot be generalized. However, the corollary to Theorem 3.9 states that if T ϵ <i>Pi> with σ(T) ≤ Gb>ob>, then T has a non-trivial lattice of invariant subspaces. After the completion of most of the work on this thesis, Stampfli [30, 31] published a proof that a paranormal operator T with σ(T) ≤ Gb>ob> is normal. His proof uses some rather deep results concerning numerical ranges whereas the proof of Theorem 3.5 uses relatively elementary methods.

Some generalizations of commutativity for linear transformations on a finite dimensional vector space

Let <i>Li> be the algebra of all linear transformations on an n-dimensional vector space V over a field <i>Fi> and let A, B, Ɛ<i>Li>. Let Ab>i+1b> = Ab>ib>B - BAb>ib>, i = 0, 1, 2,…, with A = Ab>ob>. Let fb>kb> (A, B; σ) = Ab>2K+1b> - ^σ1^A2K-1 ^{+ σ}2^A2K-3 -… +(-1)^Kσb>Kb>Ab>1b> where σ = (σb>1b>, σb>2b>,…, σb>Kb>), σb>ib> belong to <i>Fi> and K = k(k-1)/2. Taussky and Wielandt [Proc. Amer. Math. Soc., 13(1962), 732-735] showed that fb>nb>(A, B; σ) = 0 if σb>ib> is the i^th elementary symmetric function of (βb>4b>- βb>sb>)², 1 ≤ r ˂ s ≤ n, i = 1, 2, …, N, with N = n(n-1)/2, where βb>4b> are the characteristic roots of B. In this thesis we discuss relations involving fb>kb>(X, Y; σ) where X, Y Ɛ <i>Li> and 1 ≤ k ˂ n. We show: 1. If <i>Fi> is infinite and if for each X Ɛ <i>Li> there exists σ so that fb>kb>(A, X; σ) = 0 where 1 ≤ k ˂ n, then A is a scalar transformation. 2. If <i>Fi> is algebraically closed, a necessary and sufficient condition that there exists a basis of V with respect to which the matrices of A and B are both in block upper triangular form, where the blocks on the diagonals are either one- or two-dimensional, is that certain products Xb>1b>, Xb>2b>…Xb>rb> belong to the radical of the algebra generated by A and B over <i>Fi>, where Xb>ib> has the form fb>2b>(A, P(A,B); σ), for all polynomials P(x, y). We partially generalize this to the case where the blocks have dimensions ≤ k. 3. If A and B generate <i>Li>, if the characteristic of <i>Fi> does not divide n and if there exists σ so that fb>kb>(A, B; σ) = 0, for some k with 1 ≤ k ˂ n, then the characteristic roots of B belong to the splitting field of gb>kb>(w; σ) = w^2K+1 - σb>1b>w^2K-1 + σb>2b>w^2K-3 - …. +(-1)^K σb>Kb>w over <i>Fi>. We use this result to prove a theorem involving a generalized form of property L [cf. Motzkin and Taussky, Trans. Amer. Math. Soc., 73(1952), 108-114]. 4. Also we give mild generalizations of results of McCoy [Amer. Math. Soc. Bull., 42(1936), 592-600] and Drazin [Proc. London Math. Soc., 1(1951), 222-231].

Structure theorems for noncommutative complete local rings

If R is a ring with identity, let N(R) denote the Jacobson radical of R. R is local if R/N(R) is an artinian simple ring and ∩N(R)ⁱ = 0. It is known that if R is complete in the N(R)-adic topology then R is equal to (B)b>nb>, the full n by n matrix ring over B where E/N(E) is a division ring. The main results of the thesis deal with the structure of such rings B. In fact we have the following.

If B is a complete local algebra over F where B/N(B) is a finite dimensional normal extension of F and N(B) is finitely generated as a left ideal by k elements, then there exist automorphisms gb>ib>,...,gb>kb> of B/N(B) over F such that B is a homomorphic image of B/N[[xb>1b>,…,xb>kb>;gb>1b>,…,gb>kb>]] the power series ring over B/N(B) in noncommuting indeterminates xb>ib>, where xb>ib>b = gb>ib>(b)xb>ib> for all b ϵ B/N.

Another theorem generalizes this result to complete local rings which have suitable commutative subrings. As a corollary of this we have the following. Let B be a complete local ring with B/N(B) a finite field. If N(B) is finitely generated as a left ideal by k elements then there exist automorphisms gb>1b>,…,gb>kb> of a v-ring V such that B is a homomorphic image of V [[xb>1b>,…,xb>kb>;gb>1b>,…,gb>kb>]].

In both these results it is essential to know the structure of N(B) as a two sided module over a suitable subring of B.