Banach Algebra of Bounded Complex Linear Operators

The terminology and notation used here are introduced in the following articles: [18], [8], [20], [5], [7], [6], [3], [1], [17], [13], [19], [14], [2], [4], [15], [10], [11], [9], and [12]. One can prove the following propositions: (1) Let X, Y , Z be complex linear spaces, f be a linear operator from X into Y , and g be a linear operator from Y into Z. Then g · f is a linear operator from X into Z. (2) Let X, Y , Z be complex normed spaces, f be a bounded linear operator from X into Y , and g be a bounded linear operator from Y into Z. Then (i) g · f is a bounded linear operator from X into Z, and (ii) for every vector x of X holds ‖(g ·f)(x)‖ ¬ (BdLinOpsNorm(Y, Z))(g) · (BdLinOpsNorm(X,Y ))(f) · ‖x‖ and (BdLinOpsNorm(X, Z))(g · f) ¬ (BdLinOpsNorm(Y, Z))(g) · (BdLinOpsNorm(X, Y ))(f). Let X be a complex normed space and let f , g be bounded linear operators from X into X. Then g · f is a bounded linear operator from X into X. Let X be a complex normed space and let f , g be elements of BdLinOps(X, X). The functor f + g yields an element of BdLinOps(X, X) and is defined by: (Def. 1) f + g = (Add (BdLinOps(X,X),CVSpLinOps(X, X)))(f, g). Let X be a complex normed space and let f , g be elements of BdLinOps(X, X). The functor g · f yields an element of BdLinOps(X, X) and is defined as follows: