"She thanked men,—good! but thanked Somehow—I know not how—as if she ranked My gift of a nine-hundred-years-old nameWith anybody’s gift."
"Oh sir, she smiled, no doubt, Whene’er I passed her; but who passed without Much the same smile? This grew; I gave commands; Then all smiles stopped together."
It is clear from the first quote that the Duke valued his nine-hundred-years-old name, his "gift" to the Dutchess, as something that placed him above other men ... or at least should have placed him among other men in here sight. The second quote shows his supreme arrogance and possessiveness. Yes, she smiled when he passed; but all others who passed, who had not given her his "name," received the same smile. This clearly was a violation of his "specialness." If he didn't get the "special" smile, then they would all cease.
There's arrogance every place you look.
"But to myself they turned (since none puts by
The curtain I have drawn for you, but I)
And seemed as they would ask me, if they durst,
How such a glance came there;" (9-12)
These lines reveal the speaker's sense of self importance. When discussing the painting, he mentions that others have come to him as an authority.