In The Merchant Of Venice, what is the duke's opinion of the matter between Antonio and Shylock?

The duke is clearly compassionate. He feels pity for Antonio and states the following when he appears before him:

I am sorry for thee: thou art come to answer
A stony adversary, an inhuman wretch
uncapable of pity, void and empty
From any dram of mercy.

In this statement the duke calls Shylock 'a stony adversary' meaning that he has no feelings and is a tough opponent who is not willing to be swayed by any means. Shylock, he says, is pitiless and without even a smidgen of compassion. The implication is, therefore, that Antonio should expect the worst since Shylock will not back down, in any way, from his claim for restitution. Antonio acknowledges that he has learnt of the duke's numerous attempts to sway Shylock so that he may show clemency, but the Jew stubbornly refused to be turned.

Throughout the trial, the duke constantly appeals to Shylock to display some charity. He asks him to be merciful in the context of Antonio's unfortunate circumstances. When Shylock appears before him he addresses him, saying:

Shylock, the world thinks, and I think so too,
That thou but lead'st this fashion of thy malice
To the last hour of act; and then 'tis thought
Thou'lt show thy mercy and remorse more strange
Than is thy strange apparent cruelty;
And where thou now exact'st the penalty,
Which is a pound of this poor merchant's flesh,
Thou wilt not only loose the forfeiture,
But, touch'd with human gentleness and love,
Forgive a moiety of the principal;
Glancing an eye of pity on his losses,
That have of late so huddled on his back,
Enow to press a royal merchant down
And pluck commiseration of his state
From brassy bosoms and rough hearts of flint,
From stubborn Turks and Tartars, never train'd
To offices of tender courtesy.
We all expect a gentle answer, Jew.

The duke's appeal is quite clear: he believes (as do others, he says) that Shylock is pressing ahead with his demand until just before sentence is executed when he would then extend mercy and free Antonio from his claim. He says that Shylock's demand is cruel and if he should have the penalty carried out he would lose the forfeited amount. He beseeches him to show gentleness and love and seek only a portion of the principal amount owed to him out of pity for Antonio's losses which have put him at a great disadvantage. The duke uses metaphoric language to urge Shylock to draw pity from a hardened heart and display courtesy. He asks Shylock to give a gentle response.

Shylock, however, is not moved and demands the 'due and forfeit of my bond.' He refuses to accept any form of settlement. The duke then asks him how he would expect to have mercy if he had none. Shylock dismisses this contention, saying that he is only seeking justice and has not committed any crime. The duke is clearly upset by Shylock's recalcitrance and states that he may dismiss the court (and thus Shylock's claim), but he will give a learned lawyer, Bellario, an opportunity to address the court to finally settle the matter. 

Once Portia (appearing as Balthasar in disguise) had made representations to the court and determined Shylock's guilt in his malevolent desire to harm a citizen of the state, the duke pronounces the following sentence upon him:

That thou shalt see the difference of our spirits,
I pardon thee thy life before thou ask it:
For half thy wealth, it is Antonio's;
The other half comes to the general state,
Which humbleness may drive unto a fine.

The sentence is obviously used as a lesson in forgiveness. The duke makes it clear that there is a difference in how the court acts and what Shylock demanded. The duke spares him his life and awards half of his estate to Antonio, whilst the other half is forfeited to the state in the form of a fine. Furthermore, the duke orders that Shylock meets Antonio's demand that he become a Christian and donate his estate to his daughter and his son-in-law in the form of a will. If he should not, he will withdraw the pardon he had issued.