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Let $F$ be an algebraically closed field of characteristic different from$2$. We show that the images of multilinear $*$-polynomials on $UT_2$ arehomogeneous vector spaces. An analogous result holds for $UT_3$ endowed withnon-trivial grading. We further show that these results are optimal, in thefollowing sense: there exist multilinear $+$graded polynomials whose image on$UT_n$ $(n\geq 3)$ with the trivial grading is not a vector space, and whoseimage on $(UT_n)$ $(n\geq 4)$ with the $\mathbb{Z}_n$-grading is also not avector space. In particular, an analog of the L'vov-Kaplansky conjecture cannot be expected in the setting of algebras with (graded) involutions.

Values of multilinear graded $*$-polynomials on upper triangular matrices of small dimension