Murphy’s Works: Operator Calculus
Overview of Murphy’s Analytic Operators
In 1837, Murphy published a paper called “First Memoir on the Theory of Analytic Operations.” The purpose of Murphy’s paper was to present the theory of operators within calculus. Although the paper was titled the “First Memoir,” there were no followup papers.
Murphy began his paper by presenting key terminology and notation. For example, he introduced the term subjects to refer to objects on which operations are performed. Also, Murphy used the notation \([f(x)]\psi\) when discussing the operator \(\psi\) applied to the function \(f(x),\) where the subject is contained within brackets.
He then discussed special types of operators. He considered the operators \(p\) and \(q\) as fixed or free, where “in the first case a change in the order in which they are to be performed would affect the result, in the second case it would not do so” [Murphy 1837b, p. 181]. Furthermore, Murphy introduced the idea of a linear operator. According to Murphy [1837], “Linear operations in analysis are those of which the action on any subject is made up by the several actions on the parts, connected by the sign \(+\) or \(−,\) of which the subject is composed” [p. 181]. For example, let \(a\) and \(b\) be subjects and \(p\) be the operation of multiplying by the quantity \(p.\) Then \[\left[a \pm b\right]p = \left[a\right]p \pm \left[b\right]p,\] which is a linear operator according to Murphy’s definition. Interestingly, Murphy was the first mathematician to use the term linear when referring to a special class of operators [Allaire and Bradley 2002].

Figure 6. Portrait of George Peacock (Source: Public domain) 
Contributions to Linear Operator Theory
Murphy expanded on previous work in the theory of linear operators, such as the research conducted by Servois. In his elaboration on the theory of linear operators, Murphy introduced the term appendage and defined the appendage of a linear operator to be “the result of its action on zero” [Murphy 1837b, p. 188]. Here, “action” means the inverse image of the operator. The appendage is analogous to the modern day kernel of a linear transformation, and this was the first time that the kernel of an operator had been examined [Allaire and Bradley 2002].
Murphy further expanded on his concept of the appendage. For example, he considered \([P]\theta=0,\) where \(\theta\) is a linear operator and \(P\) the subject connected with the nature of the operator. He then stated that \(P\) would express a form. Here, the form will be the set of all elements which are mapped to \(0\) by \(\theta.\) It is reasonable to conjecture that Murphy adopted this term from the 1830 work, A Treatise on Algebra, by George Peacock (17911858).
Finally, Murphy considered the equation \(\theta \iota = \iota x,\) where \(\theta,\) \(\iota,\) and \(x\) are operators given without a subject and a general identity. He then defined \(\iota\) to be intermediate with respect to \(\theta\) and \(x\). If you are given \(\theta\) or \(x\) and the intermediate \(\iota\), then you can solve for the third operation; i.e., \(\theta = \iota x \iota^{1}\) and \(x = \iota ^{1} \theta \iota\). In modern algebra we would call \(x\) the conjugate of \(\theta\). Murphy [1837b] demonstrated that intermediate operators are also intermediate between any operations that are the same functions of the extremes (here, the extremes are \(\theta\) and \(x\)). Murphy accomplished this by demonstrating that if \(\theta i = i x\), then \(\theta ^{n} i = i x^{n}\). Essentially, Murphy examined the closure property of intermediate operations.
Throughout his paper, Murphy derived several results involving differentials, such as a method for expanding a function into a series. In his derivation, he created expansions that are equivalent to the “theorem of Taylor” [1837b, p. 183]. In addition, he discovered some “remarkable properties” of the operation \(\psi\) which changes \(x\) into \(x + h\). Louis François Antoine Arbogast (17591803) called this operator the varied state [1800] and Murphy stated that this operator was free and linear.
Murphy then incorporated his concept of a “free” linear operator (the modern definition of a linear operator) into the calculus  an important contribution to the general theory of operators. Despite its significance, this paper was Murphy’s first and only work in the category of operator calculus.