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.
. . . if there be among the gifts of God none greater than love, and there is no greater gift of God than the Holy Spirit, what follows more naturally than that He is Himself love, who is called both God and of God? And if the love by which the Father loves the Son, and the Son loves the Father, ineffably demonstrates the communion of both, what is more suitable than that He should be specially called love, who is the Spirit common to both? For this is the sounder thing both to believe and to understand, that the Holy Spirit is not alone love in that Trinity, yet is not specially called love to no purpose, for the reasons we have alleged; just as He is not alone in that Trinity either a Spirit or holy, since both the Father is a Spirit, and the Son is a Spirit; and both the Father is holy, and the Son is holy,—as piety doubts not. And yet it is not to no purpose that He is specially called the Holy Spirit; for because He is common to both, He is specially called that which both are in common. (On the Trinity, 15.19.37)
The Holy Spirit is the act of God between the Father and the Son infinitely loving and delighting in each other.
It appears by the holy Scriptures, that the Holy Spirit is the perfect act of God. . . . And it also appears that the Holy Spirit is this act of the Deity, even love and delight, because from eternity there was no other act in God but thus acting with respect to himself, and delighting perfectly and infinitely in himself, or that infinite delight there is between the Father and the Son; for the object of God’s perfect act must necessarily be himself, because there is no other.
I’m not fully satisfied with any of these three expressions, but I can’t help but appreciate their attempt to relate Father, Son, and Spirit in a way that accords with the weight that Scripture gives to each one’s personality.