Citation
Gaines, Fergus John (1966) Some generalizations of commutativity for linear transformations on a finite dimensional vector space. Dissertation (Ph.D.), California Institute of Technology. http://resolver.caltech.edu/CaltechTHESIS:09222015114033548
Abstract
Let L be the algebra of all linear transformations on an ndimensional vector space V over a field F and let A, B, ƐL. Let A_{i+1} = A_{i}B  BA_{i}, i = 0, 1, 2,…, with A = A_{o}. Let f_{k} (A, B; σ) = A_{2K+1}  ^{σ}1^{A}2K1 ^{+} ^{σ}2^{A}2K3 … +(1)^{K}σ_{K}A_{1} where σ = (σ_{1}, σ_{2},…, σ_{K}), σ_{i} belong to F and K = k(k1)/2. Taussky and Wielandt [Proc. Amer. Math. Soc., 13(1962), 732735] showed that f_{n}(A, B; σ) = 0 if σ_{i} is the i^{th} elementary symmetric function of (β_{4} β_{s})^{2}, 1 ≤ r ˂ s ≤ n, i = 1, 2, …, N, with N = n(n1)/2, where β_{4} are the characteristic roots of B. In this thesis we discuss relations involving f_{k}(X, Y; σ) where X, Y Ɛ L and 1 ≤ k ˂ n. We show: 1. If F is infinite and if for each X Ɛ L there exists σ so that f_{k}(A, X; σ) = 0 where 1 ≤ k ˂ n, then A is a scalar transformation. 2. If F 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 twodimensional, is that certain products X_{1}, X_{2}…X_{r} belong to the radical of the algebra generated by A and B over F, where X_{i} has the form f_{2}(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 L, if the characteristic of F does not divide n and if there exists σ so that f_{k}(A, B; σ) = 0, for some k with 1 ≤ k ˂ n, then the characteristic roots of B belong to the splitting field of g_{k}(w; σ) = w^{2K+1}  σ_{1}w^{2K1} + σ_{2}w^{2K3}  …. +(1)^{K} σ_{K}w over F. 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), 108114]. 4. Also we give mild generalizations of results of McCoy [Amer. Math. Soc. Bull., 42(1936), 592600] and Drazin [Proc. London Math. Soc., 1(1951), 222231].
Item Type:  Thesis (Dissertation (Ph.D.)) 

Subject Keywords:  Mathematics 
Degree Grantor:  California Institute of Technology 
Division:  Physics, Mathematics and Astronomy 
Major Option:  Mathematics 
Thesis Availability:  Public (worldwide access) 
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Defense Date:  4 April 1966 
Record Number:  CaltechTHESIS:09222015114033548 
Persistent URL:  http://resolver.caltech.edu/CaltechTHESIS:09222015114033548 
Default Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided. 
ID Code:  9165 
Collection:  CaltechTHESIS 
Deposited By:  Leslie Granillo 
Deposited On:  25 Sep 2015 16:46 
Last Modified:  25 Sep 2015 16:46 
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