A number of Ukrainian classical musicians who have always been considered Russian were actually from Ukraine. The following are particularly important examples:

Legendary pianists:

Vladimir Horowitz, 1903-1989, was born in Kyiv. He began study at the Kyiv Conservatory in 1912, and gave his first solo recital in Kharkiv in 1920. After the revolution, he emigrated to the West in 1925, where he immediately embarked on his career as a concert pianist. He would soon be an international superstar. He remained in the West for the rest of his life, giving hundreds of concerts and making many recordings. As relations improved between Russia and the United States, he returned to Russia to give some concerts in 1986. A memorable one from Moscow was broadcast to an international audience. Despite the fact that the Soviets did not publicize it at all, it was sold out quickly and created a sensation. It was a touching triumph for this great pianist not long before his death. 

Sviatoslav Richter, 1915-1997
Born in Zhytomyr, Ukraine to a musical family, he and his family moved to Odessa, Ukraine in 1921, where he received significant early training. In 1937, he moved to Moscow to study piano at the Moscow Conservatory. He went on to spend his entire career in the Soviet Union, and gained an international reputation by the 1950s. He was allowed to tour outside of Russia, and made many recordings with famous orchestras in various countries. 

Emil Gilels, 1916-1985, was born in Odessa, Ukraine and received an excellent musical education there. In 1929, he was accepted at the Odessa Conservatory and gave his first public concert. In 1936, he began to study at the Moscow Conservatory and won an important musical competition. This brought him wide fame. He stayed in Russia for the rest of his life. He had a lengthy, brilliant career as a concert artist, recording artist, and teacher. In 1958 he chaired the jury for the inaugural International Tchaikovsky Competition in Moscow which awarded first prize to Van Cliburn.  

A Legendary Singer:

Ivan Kozlovsky, 1900-1993
This famous tenor was born in Marianivka, Ukraine. He studied in Kyiv and sang operatic roles at Poltava and Kharkiv in the early 1920s. He soon went on to be a leading lyric tenor at the Bolshoi Theatre in Moscow. He gave many concerts throughout the Soviet Union, and gained great fame. Late in his life, he verified the story that he had been Stalin's favorite singer. Because his brother had emigrated to the West, he was never allowed to tour outside of Russia, but fortunately, many of his recordings are widely available by now. Throughout his career he actively promoted Ukrainian music and made many recordings of it. In 1970, he funded a music school in Marianivka. 

A Legendary Violinist:

David Oistrakh, 1908-1974
This famous violinist was born in Odessa, Ukraine and studied at the Odessa Conservatory. In 1927 he made his concert debut in Kyiv. He moved to Moscow that year, where he became a professor at the Moscow Conservatory. He was associated with famous composers such as Shostakovich, Khachaturian, and Prokofiev, and premiered several of their works. Allowed to tour in the West, he achieved widespread international fame.