Here’s Karl Barth’s answer:
. . . even if the Father and the Son might be called “person” (in the modern sense of the term), the Holy Spirit could not possibly be regarded as the third “person.” In a particularly clear way the Holy Spirit is what the Father and the Son also are. He is not a third spiritual Subject, a third I, a third Lord side by side with two others. He is a third mode of being of the one divine Subject or Lord.
. . .
He is the common element, or, better, the fellowship, the act of communion, of the Father and the Son. He is the act in which the Father is the Father of the Son or the Speaker of the Word and the Son is the Son of the Father or the Word of the Speaker. (CD I,1, 469)
This sounds on the surface like a denial of full trinitarianism (and I am a little uncomfortable with it), but it shares much in common with the views of Augustine and Jonathan Edwards, both of whom tended to talk about the Spirit in ways that seem less than fully personal.