On Motzkin sequence spaces via <i>q</i> -analog and compact operators

Abstract We aim to develop a <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>q</m:mi> </m:math> q -analog of recently introduced Motzkin sequence spaces by Erdem et al. [ Motzkin sequence spaces and Motzkin core , Numer. Funct. Anal. Optim. 45 (2024), no. 4–6, 283–303] by using <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>q</m:mi> </m:math> q -Motzkin numbers and introduce sequence spaces <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>c</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi mathvariant="fraktur">M</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>q</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> c\left({\mathfrak{M}}\left(q)) and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mi>c</m:mi> </m:mrow> <m:mrow> <m:mn>0</m:mn> </m:mrow> </m:msub> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi mathvariant="fraktur">M</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>q</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>.</m:mo> </m:math> {c}_{0}\left({\mathfrak{M}}\left(q)). We investigate some topological properties, compute bases, and obtain their duals. For <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>X</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mo stretchy="false">{</m:mo> <m:mrow> <m:mi>c</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi mathvariant="fraktur">M</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>q</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mo>)</m:mo> </m:mrow> <m:mo>,</m:mo> <m:msub> <m:mrow> <m:mi>c</m:mi> </m:mrow> <m:mrow> <m:mn>0</m:mn> </m:mrow> </m:msub> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi mathvariant="fraktur">M</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>q</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mo stretchy="false">}</m:mo> </m:mrow> </m:math> X\in \{c\left({\mathfrak{M}}\left(q)),{c}_{0}\left({\mathfrak{M}}\left(q))\} and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>Y</m:mi> <m:mo>∈</m:mo> <m:mrow> <m:mo stretchy="false">{</m:mo> <m:mrow> <m:msub> <m:mrow> <m:mi>ℓ</m:mi> </m:mrow> <m:mrow> <m:mi>∞</m:mi> </m:mrow> </m:msub> <m:mo>,</m:mo> <m:mi>c</m:mi> <m:mo>,</m:mo> <m:msub> <m:mrow> <m:mi>c</m:mi> </m:mrow> <m:mrow> <m:mn>0</m:mn> </m:mrow> </m:msub> <m:mo>,</m:mo> <m:msub> <m:mrow> <m:mi>ℓ</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> </m:mrow> </m:msub> </m:mrow> <m:mo stretchy="false">}</m:mo> </m:mrow> <m:mo>,</m:mo> </m:math> Y\in \{{\ell }_{\infty },c,{c}_{0},{\ell }_{1}\}, some results pertaining to characterization of matrix class <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>X</m:mi> <m:mo>,</m:mo> <m:mi>Y</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> \left(X,Y)