On Banach subalgebras of the Dirichlet Hardy algebra $$\mathscr {H}^\infty $$ consisting of lacunary Dirichlet series

Abstract Let $$\mathscr {H}^\infty $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>H</mml:mi> <mml:mi>∞</mml:mi> </mml:msup> </mml:math> be the set of all Dirichlet series $$\textstyle f\!=\!{{\sum \limits _{n=1}^\infty }} a_nn^{-s}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mstyle> <mml:mrow> <mml:mi>f</mml:mi> <mml:mspace/> <mml:mo>=</mml:mo> <mml:mspace/> <mml:mrow> <mml:munderover> <mml:mo>∑</mml:mo> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>