Greek (THGNT): ὃς ὢν ἀπαύγασμα τῆς δόξης καὶ χαρακτὴρ τῆς ὑποστάσεως αὐτοῦ, φέρων τε τὰ πάντα τῷ ῥήματι τῆς δυνάμεως αὐτοῦ, καθαρισμὸν τῶν ἁμαρτιῶν ποιησάμενος ἐκάθισεν ἐν δεξιᾷ τῆς μεγαλωσύνης ἐν ὑψηλοῖς
ὃς-a relative pronoun (see Hebrews 1:2); translate "he" or "who." This Greek form is masculine nominative singular. Jody A. Bernard argues that Hebrews 1:3-4 depends grammatically on Hebrews 1:1-2 and refers back to the noun "Son" (Bernard, The Mysticism of Hebrews, page 132).
ὢν-nominative singular masculine present participle of εἰμί: translate "is" or "being." It is the first of three participles "that describe the person and work of the Son" (Harris, Hebrews); B.F. Wescott opines that ὢν rules out adoptionism "and affirms the permanence of the divine essence of the Son during his historic work" (The Epistle to the Hebrews, page 9). That seems to be asking a Greek present participle to bear a heavy load which it cannot sustain.
ἀπαύγασμα τῆς δόξης-ἀπαύγασμα is nominative singular neuter; the word occurs 1x in the GNT. LSJ gives this definition: "radiance, effulgence, of light beaming from a luminous body." See Wisdom 7:26 and compare Abbott-Smith Manual Lexicon, page 45. Understood passively, which is the most probable meaning, ἀπαύγασμα would denote "reflection." See Louw-Nida 14.48. Ken Schenck argues for a passive sense of ἀπαύγασμα at https://kenschenck.blogspot.com/2021/06/hebrews-11-4-explanatory-notes.html
Westcott's analysis of ἀπαύγασμα is not that helpful for me since he focuses more on lexical semantics and other features of Hebrews, and his commentary is heavily colored by theological impositions. For instance, he imposes the notion of homoousios upon ἀπαύγασμα and χαρακτὴρ. On the other hand, Attridge writes:
The context of Hebrews itself, where ἀπαύγασμα is paralleled with "imprint" (χαρακτὴρ), may support a passive understanding of ἀπαύγασμα, although that second term is not entirely free from ambiguity. On the other hand, the parallelism may not be synonymous, but antithetical, as in the two preceding clauses. In such poetic language complete precision is not to be expected. The image, in whatever sense it may be taken, serves, like the following, to affirm the intimate relationship between the Father and the pre-existent Son, through whom redemption is effected.τῆς δόξης is part of a genitive chain: the construction is potentially an objective genitive signifying that the Son expresses the Father's radiance (see David L. Allen), but Rogers and Rogers also note that the genitive of content is possible here. The mention of "glory" in Hebrews likely depends on the OT word, kabod, which the LXX translates with doxa (Allen).
Harris (Hebrews) posits that "Δόξης is likely a descriptive gen. (Wallace 79–81), although it is possibly a gen. of quality or attrib. gen. (R 496; 'glorious radiance'). Thus the Son perfectly radiates God’s own glory."
καὶ χαρακτὴρ τῆς ὑποστάσεως αὐτοῦ, φέρων τε τὰ πάντα τῷ ῥήματι τῆς δυνάμεως αὐτοῦ-Allen contends that both ὢν and φέρων (present participles) "should be construed as concessive adverbial modifiers" of ποιησάμενος. He observes that these participles make affirmations about the Son of God (mentioned in 1:2) although they're adverbial rather than adjectival. William L. Lane concurs with Allen's assessment in his WB commentary.
See Allen, David L. Hebrews: 35 (New American Commentary) (Kindle Location 3299). B&H Publishing. Kindle Edition.
ἀπαύγασμα and χαρακτὴρ predicate similar functions of the Son since they occur in synonymously parallel fashion (Lane), but Harold Attridge offers the rejoinder that the parallelism might be antithetical. In any event, Lane adds (Hebrews 1-8, WB Commentary):
The term χαρακτήρ occurs only here in the NT. It is found three times in the LXX, but in a nontechnical sense (Lev 13:28; 2 Macc 4:10; 4 Macc 15:4; cf. T. Sim. 5:5). Philo shows a strong preference for this term, which occurs fifty-one times in his works with a variety of associated meanings. He applies it frequently to man, whose soul bears the imprint (χαρακτήρ) of God (Allegorical Interpretation 3.95; On the Virtues 52; The Worse Attacks the Better 83). But if the soul may be compared to the image of the invisible God bearing the engraving of the divine seal, the stamp of that seal is the eternal Logos (ἧς ὁ χαρακτήρ ἐστιν ὁ αἴδιος λόγος On Noah’s Work as a Planter 18). It is imperative, however, not to exaggerate the importance of such alleged “parallels” to the expression in v 3. In his work On the Unchangeableness of God (55), Philo states that God’s being is “simple” (ψιλὴν τὴν ὕπαρξιν), “without other definite characteristic” (ἄνευ χαρακτῆρος).
χαρακτὴρ τῆς ὑποστάσεως αὐτοῦ-Rogers and Rogers favor the genitive of description for this portion of 1:3 (i.e., "the exact representation of his substance"). But as Lane, Allen, Ellingworth inter alia point out, χαρακτὴρ is tricky to define and so is ὑπόστασις--particularly when the latter term has been shaped and understood through the lens of fourth-century debates about Christology. See Attridge, Hebrews, page 44; Moses Stuart, A Commentary on the Epistle to the Hebrews, page 269; James Moffatt, Epistle to the Hebrews, page 6; Westcott, The Epistle to the Hebrews, page 13. Stuart opts for the classical denotation of ὑπόστασις pre-Nicaea, that is, substance or essence, not person. See Jeremiah 10:17.
The rendering "substance" is possibly not the best way to translate ὑπόστασις in Hebrews 1:3. BDAG Greek-English Lexicon gives this observation for ὑπόστασις:
"the essential or basic structure/nature of an entity, substantial nature, essence, actual being, reality . . ."
BDAG glosses Heb. 1:3 this way:
"a(n) exact representation of (God's) real being (i.e. as one who is in charge of the universe)"
φέρων is present active participle of φέρω, meaning "carry, bear, bring forth."
φέρων τε τὰ πάντα τῷ ῥήματι τῆς δυνάμεως αὐτοῦ-Attridge identifies the construction as a descriptive genitive, which bears possible marks of Semitic influence via the LXX. The subject of this passage is apparently the Son (Christ) as Westcott explains; LEB translates: "sustaining all things by the word of power" but NET renders "and he sustains all things by his powerful word"; See Deuteronomy 1:9.
Postpositive τε-"and" or "but." τε suggests a change in the relation spoken about in the discourse (Westcott, page 13): its purpose is likely adjungit rather than conjungit (ibid.). Mounce defines the particle this way: enclitic, can function as a conj., serving either as a lightly-appending link, Acts 1:15; and, Acts 2:3; or as an inclusive prefix, Lk. 12:45; both, Lk. 24:20; Acts 26:16.
τὰ πάντα ("all things" or "the universe")-for a comparison, see Romans 11:36; Colossians 1:17; Hebrews 2:10; Revelation 4:11.
τῷ ῥήματι is dative singular neuter of ῥῆμα ("word") and it could be an instrumental dative (see Rogers and Rogers), which would yield the sense "by means of His word." Westcott points to Genesis 1:3 (LXX) and Hebrews 11:3: he finds significance in the use of ῥῆμα over against the occurrence of λόγος as we see in John 1:1-3, but the term refers to the Son's utterance here.
Bill Mounce: δυνάμεως is clearly an Hebraic genitive and hence the NLT translates,
“he sustains everything by the mighty power of his command.” A “literal”
translation would produce a meaningless phrase if all it did was
Albert J. Coetsee reckons that τῆς δυνάμεως is a genitive of quality, that is, "a Hebraism where the genitive functions as an adjective (cf. Black 1987:190; Blass & Debrunner 1961:§165; Cockerill 2012:95)." So he favors the translation, "by his powerful word," and Coetsee infers that the Son is Creator and Sustainer of the cosmos like a number of other commentators.
καθαρισμὸν τῶν ἁμαρτιῶν ποιησάμενος ἐκάθισεν ἐν δεξιᾷ τῆς μεγαλωσύνης ἐν ὑψηλοῖς:
καθαρισμὸν ("purification") is a noun whose form is accusative singular masculine. Cf. 2 Exodus 30:10; Peter 1:9. ποιησάμενος is aorist participle middle of ποιέω: translate "having made." Zerwick-Grosvenor explain that the participle along with its object communicate the same idea as καθαρίσας (the cognate verb of καθαρισμός). See Zerwick-Grosvenor, A Grammatical Analysis, page 654. For Harris (Hebrews), ποιησάμενος is temporal ("after making purification") or causal ("because he made purification").
τῶν ἁμαρτιῶν (genitive plural feminine) is employed "of sins generally" (Westcott, Hebrews, page 15): John 1:29 is supposed to provide a contrast with τὴν ἁμαρτίαν. The action signified by ποιησάμενος also occurs before the Son ἐκάθισεν ἐν δεξιᾷ τῆς μεγαλωσύνης ἐν ὑψηλοῖς.
ἐκάθισεν is the first aorist active of καθίζω; "make one sit" or when used intransitively, "take one's seat, sit down" (Zerwick-Grosvenor, page 654). I agree with Benjamin J. Ribbens that the purification happens in the heavenly sanctuary (the greater Most Holy): note the use of ἐν ὑψηλοῖς and compare Hebrews 10:11-13. See 4.3.1 of Ribbens. As Philip Church writes, καθαρισμὸν τῶν ἁμαρτιῶν ποιησάμενος adverbially modifies ἐκάθισεν (Hebrews and the Temple, page 283).
ἐν δεξιᾷ is a prepositional phrase coupled with the dative singular feminine of δεξιός: "at/in the right hand [of the Majesty]" (Psalm 16:8, 11; 110:1; 1 Chronicles 6:39; Isaiah 44:20; 63:12; Acts 2:33; 5:31; Romans 8:34; Ephesians 1:20; Colossians 3:1; 1 Peter 3:22; Hebrews 1:13; 8:1; 10:12-13; 12:2). Most scholars insist that the locative statement "at the right hand" should not be understood locally or spatially: "It denotes entrance into a position of divine authority" (Moffatt, Epistle to the Hebrews, page 8).
τῆς μεγαλωσύνης ἐν ὑψηλοῖ is apparently a circumlocution for God (YHWH or the Father): " 'The Majesty' expresses the idea of God in His greatness" (Westcott, Hebrews, page 16) and Philip Church maintains that "μεγαλωσύνη seems to be a reverential periphrasis for God" (Hebrews and the Temple, page 282). Compare Buchanan, Hebrews, page 8 and Hebrews 8:1; Jude 25; 1 Clement 16:2; 17:4; 36:2; 58:1. For LXX examples, see Deuteronomy 32:3; Psalm 79:11 (78:11 LXX); 145:3 (144:3 LXX). Moffatt uses similar language as Church in Epistle to the Hebrews, page 8.
τῆς μεγαλωσύνης is genitive singular feminine of μεγαλωσύνη and the morphological ending tells us the noun is abstract as opposed to being concrete (compare δικαιοσύνη and related nouns). August Tholuck maintains that the abstract is more prominent than the concrete in Hebrews 1:3, just as "the Majesty" (an abstraction) is apparently more emphatic than "the majestic" which is concrete (Tholuck, A Commentary on the Epistle to the Hebrews, page 132); however, Henry Alford suggests that one should drop the article in English translation to make "Majesty/majesty" more emphatic in terms of the abstract and concrete distinction. See Alford, The Greek Testament, page 4:146; Moses Stuart, A Commentary on the Epistle to the Hebrews, page 287.
ἐν ὑψηλοῖ-another prepositional phrase coupled with the dative case that could be rendered "in the highest" or "in the heights." Again, the phrase is likely not to be understood spatially, but probably is a metaphorical utterance. Moffatt thinks the phrase should go grammatically with ἐκάθισεν (cf. Hebrews 8:1), but the Expositor's Greek Testament proffers that ἐν ὑψηλοῖ should go with μεγαλωσύνης. Henry Alford indicates that the reading in the EGT is more natural, but compare Westcott, op. cit., page 16.