Grace and peace to you from him who is and who was and who is coming, and from the seven spirits that are before his throne, and from Jesus the Christ...
When John refers to 'him who is and who was and who is coming' throughout the Revelation, it is an epithet he reserves for God, 'the Father'. And of course, 'Jesus the Christ' refers to Jesus.
Since we have the Father and the son, readers coming from a trinitarian position expect any third persons to be identified with the holy spirit. For the third person, John does use the word 'spirit', but it is puzzling he actually refers to seven spirits, ἑπτὰ πνεύματων.
A few early interpreters pointed to Isaiah 11.2. Some English translators follow this interpretation, and so render the Greek as 'the sevenfold Spirit' (e.g. Amplified, Complete Jewish Bible, Living Bible, New Living Translation). However, this is an interpretive gloss done to satisfy theological concerns.
The Greek means 'seven spirits'.
Four layers of symbolism
If we track references to these 'seven spirits' in the Revelation, we only find them mentioned three more times: in Jesus' possession alongside the 'seven stars' (3.4), symbolized as 'seven lamps of fire' (4.5), and again symbolized as the 'seven eyes' of the Lamb (5.6).
It should be evident that these 'seven spirits' have multiple layers of symbolism attached to them. Each layer needs to be investigated to really get a hold on John's intended meaning when he writes about the 'seven spirits'.
1. Seven stars
Revelation 3.4 mentions the seven spirits alongside the 'seven stars' as things Jesus possesses. The 'seven stars' were explained to be symbols for seven angels that stand on behalf of the seven churches in Asia (Revelation 1.20).
Following this introductory passage, the 'seven angels' are next seen receiving trumpets, while another angel brings an incense offering to God, which includes the saints' prayers (Revelation 8.1-5).
2. Seven lamps of fire
The Revelation as a whole incorporates a great deal of temple imagery, e.g. the Lamb (twenty-eight times), the altar (6.9; 8.3; etc.), the priesthood (1.6; 5.10; 20.6), the temple (3.12; 7.15; etc.), the ark (11.19), and incense (5.8; 8.3).
Nearly all of this temple imagery is centered around God's throne, which where we find the seven spirits symbolized as 'seven lamps of fire'. Where we saw the seven angels symbolized as 'seven stars', we then found that 'seven lamps' were used to symbolize the seven churches of Asia (Revelation 1.20).
Putting these two symbols together -- seven lamps and seven lampstands -- within the context of the Jewish temple, the emergent image is that of the menorah. The seven spirits function as a 'light' within the seven churches.
3. Seven eyes
As John symbolizes Jesus as 'the Lamb', he includes a few additional symbols as well: the Lamb has seven horns ('horns' may represent anointing and power) as well as 'seven eyes, which are the seven spirits of God sent out into all the earth' (Revelation 5.6)
Here we have a deliberate reference to a passage in Zechariah concerned with the anointing of Joshua (chapter 3) and Zerubbabel (chapter 4) for restoring Jerusalem and the temple. In this prophetic vision, Zechariah sees a seven-branched menorah flanked by two olive trees. The seven lamps of the lampstand are further symbolized as 'the seven eyes of Yahweh' (4.10b; cf. 3.9) 'which range through the whole earth', nearly identical to what we have in Revelation 4-5.
The description that the seven eyes 'range through the whole earth' is similar to an earlier vision in Zechariah 1, where the prophet sees a vision of angels sent by God 'to patrol the earth'. As God commands, so the angels act on the earth.
4. Seven angels
Angelology was a common feature of Second Temple Judaism, from the Persian period onward.
Versions of 1 Enoch 17.8 and 90.20 mention 'the seven archangels' and 'those first seven white ones'. Tobit 12.15 refers to 'the seven angels who stand ready and enter before the glory of the Lord' (i.e. before God's throne); one of them even offers saintly prayers to God in the form of incense, similar to what we see in Revelation 8. The book of Jubilees has 'the angels of the presence' as a unique group of angels. The Testament of the Twelve Patriarchs 3.8 has the 'seven men in white raiment'. Lists of these 'seven angels' or 'seven men' frequently included Michael and Gabriel.
The idea continued into early Christianity: Luke 1.19 has Gabriel identify himself as 'stand[ing] in the presence of God'. Clement of Alexandria mentions 'the first-born princes of the angels ... are seven'.
An early portrayal of seven angels might be found in Ezekiel 9.2, but the idea certainly began to solidify in Jewish thought during the Persian and Hellenistic periods, as seen in the books above. This portrayal of a specific class of 'seven angels' acting as God's personal administrators may have been influenced by the court of Persian politics (e.g. Ezra 7.4).
When we consider all of the ideas described above (angels offering prayer as incense; angels patrolling the earth as extensions of God's will; a class of angels standing before God's throne; seven chief angels), we see each of them in the Revelation to one degree or another.
I suggest that the 'seven spirits which are before God's throne' should be interpreted as, essentially, 'the seven angels of God's presence' common to the time period.
In Revelation 1.4, then, John is not blessing his readers with a trinitarian formula (i.e. 'Grace and peace to you from the Father, and form the Holy Spirit, and from the Son'). Instead, John is sending blessing from heaven's royal court, from the king (him who is), the king's seven administrators (the seven spirits), and from the co-regent (Jesus).
This would be roughly equivalent to 1 Timothy 5.21, where the epistle's author issues a command on behalf 'of God and of Christ Jesus and of the elect angels'.